Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring Zprqs

In this paper, we examine the interplay between the structural and spectral properties of the unit graph G(Zn) for n=p1r1p2r2...pkrk, where p1,p2,...,pk are distinct primes and k,r1,r2,...,rk are positive integers such that at least one of the ri must be greater than 1. We first analyze the structure of the unit graph of Zn as a generalized join graph under these conditions. We then determine the Laplacian spectrum of G(Zn) and deduce that it is integral for all n. Consequently, we obtain Laplacian spectral radius and algebraic connectivity of G(Zn). We also prove that the vertex connectivity of G(Zpq) is (p−2)q, where 2≠p<q. We deduce the vertex connectivity of G(Zn) when n=prqs, where 2≠p<q are primes and r,s are positive integers. Finally, we present conjectures about the vertex connectivity of G(Zn) when n=p1p2...pk and n=p1r1p2r2...pkrk, where pi are distinct primes, ri are positive integers, and 1≤i≤k.

Keywords:

Subject:

Computer Science and Mathematics - Mathematics

MSC: 05C25; 05C50; 05C75

1. Introduction

For a positive integer n,

Z_{n}

denotes the ring of integers modulo n. In 1990, the unit graph was first introduced by Grimaldi [1] for the ring

Z_{n}

as follows: the unit graph

G (Z_{n})

is the graph obtained by setting all the elements of

Z_{n}

to be vertices and defining distinct vertices x and y to be adjacent if and only if

x + y \in U (Z_{n})

. He discussed certain basic properties of the structure of the unit graph

G (Z_{n})

and studied the degree of a vertex, covering number, independence number, Hamilton cycles, and chromatic polynomial of the graph

G (Z_{n})

. More about the unit graph

G (Z_{n})

can be seen in [2,3,4]. Later, Ashrafi et al. [5] generalized the unit graph from

G (Z_{n})

G (R)

for an arbitrary ring R. They studied the chromatic index, diameter, girth, and planarity of

G (R)

. Some of the work associated with the unit graph on the rings can be found in [6,7,8,9].

In recent years, many researchers have studied the Laplacian spectrum and vertex connectivity of graphs associated with algebraic structures. In 2020, Chattopadhyay et al. [10] studied the Laplacian spectrum of the zero divisor graph

Γ (Z_{n})

of the ring

Z_{n}

. They discussed the Laplacian integrality, algebraic connectivity, vertex connectivity, and Laplacian spectral radius of

Γ (Z_{n})

. For other related works on the Laplacian spectrum and vertex connectivity of graphs associated to the ring

Z_{n}

, one may refer to [11,12]. Shen et al. [3] determined the Laplacian spectrum of the unit graphs of the ring

Z_{n}

for

n = p^{m}

, where p is an odd prime and m is a positive integer. They proved that the algebraic connectivity and vertex connectivity of

G (Z_{n})

coincide if and only if

n = p^{m}

In this paper, we investigate the structure of

G (Z_{n})

. Based on this structure, we study the Laplacian spectrum and vertex connectivity of

G (Z_{n})

for various n. The paper is arranged as follows: In Section 2, we provide the preliminary concepts and results that are used throughout the paper. In Section 3, we examine the structure of

G (Z_{n})

for

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

p_{1}, p_{2}, . . ., p_{k}

are distinct primes and

k, r_{1}, r_{2}, . . ., r_{k}

are positive integers such that at least one of the

r_{i}

must be greater than 1. We prove that the graph

G (Z_{n})

is a generalized join of certain complete graphs and null graphs. In Section 4, we study the Laplacian spectrum of

G (Z_{n})

, we prove that

G (Z_{n})

is Laplacian integral, and we deduce the algebraic connectivity and Laplacian spectral radius of

G (Z_{n})

. In Section 5, we investigate the vertex connectivity of

G (Z_{p q})

and

G (Z_{p^{r} q^{s}})

, where

2 \neq p < q

are primes and r and s are positive integers, based on their structure and Menger’s theorem. Moreover, we present the following conjectures:

Conjecture I: Let

n = p_{1} p_{2} . . . p_{k}

, where

2 \neq p_{1} < p_{2} < . . . < p_{k}

are primes. Then, the vertex connectivity of

G (Z_{n})

(p_{1} - 2) \prod_{2 \leq i \leq k} p_{i} .

Conjecture II: Let

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

2 \neq p_{1} < p_{2} < . . . < p_{k}

are primes and

k, r_{1}, r_{2}, . . ., r_{k}

are positive integers such that at least one of the

r_{i}

must be greater than 1. Then, the vertex connectivity of

G (Z_{n})

(p_{1} - 2) p_{1}^{(r_{1} - 1)} \prod_{2 \leq i \leq k} p_{i}^{r_{i}} .

2. Preliminaries

In this section, we will present preliminary definitions and theorems that will be necessary for the following sections. Let G be a graph with vertex set

V (G) = {v_{1}, v_{2}, . . ., v_{n}}

and edge set

E (G) = {e_{1}, e_{2}, . . ., e_{m}}

. For

1 \leq i, j \leq n

, two vertices

v_{i}

and

v_{j}

in G are adjacent (or neighbors) in G if

v_{i}

and

v_{j}

are endpoints of an edge e of G, and we write

v_{i} \sim v_{j}

v_{i}

is adjacent to

v_{j}

in G. For

v \in V (G)

, we denote by

N_{G} (v)

the set of all neighbors of v in G. The degree of a vertex v in G, denoted by

d e g (v)

, is the number of edges incident with it. A path in a graph is a sequence of distinct vertices with the property that each vertex in the sequence is adjacent to the next vertex of it. The graph G is said to be connected if G contains a path between every pair of vertices. A complete graph is a graph in which each pair of distinct vertices is joined by an edge. We denote the complete graph with n vertices by

K_{n}

. The complement of

K_{n}

is a null graph and is denoted by

{\bar{K}}_{n}

. A clique of a graph G is a complete subgraph of G. A coclique in a graph G is a set of pairwise nonadjacent vertices. An isomorphism of graphs

G_{1}

and

G_{2}

G_{1} ≅ G_{2}

, is a bijection

ϕ

between the vertex sets of

G_{1}

and

G_{2}

such that for any two vertices x and y of

G_{1}

, x and y are adjacent in

G_{1}

if and only if

ϕ (x)

and

ϕ (y)

are adjacent in

G_{2}

. For two graphs

G_{1}

and

G_{2}

with disjoint vertex sets, the join

G_{1} \lor G_{2}

G_{1}

and

G_{2}

is the graph obtained from the union of

G_{1}

and

G_{2}

by adding new edges from each vertex of

G_{1}

to every vertex of

G_{2}

Let R be a ring with unity and

U (R)

be the set of units of R. The unit graph

G (R)

of R is the graph whose vertices are all the elements of R, defining distinct vertices x and y to be adjacent if and only if

x + y

is a unit in R. Let R be a commutative ring with unity. An element

r \in R

is nilpotent if there exists an integer

k > 1

such that

r^{k} = 0

. The nilradical of R, denoted

n i l (R)

, is the set of all nilpotent elements of R. An ideal

N \neq R

in R is a prime ideal if

a b \in N

implies that either

a \in N

b \in N

for

a,

b \in R

. The nilradical of R is the intersection of prime ideals of R. A maximal ideal of R is an ideal m different from R such that there is no proper ideal I of R properly containing m. The Jacobson radical

J (R)

of R is the intersection of maximal ideals of R. Every maximal ideal in R is a prime ideal. So,

J (R) = n i l (R)

. Recall that, the sum of a unit and a nilpotent element is a unit. The following two results give some properties of the unit graph

G (R)

Lemma 1.

[5] Let R be a commutative ring and suppose that

J (R)

denotes the Jacobson radical of R. If

x, y \in R

, then the following statements hold:

1.: If $x + J (R)$ and $y + J (R)$ are adjacent in $G (R / J (R))$ , then every element of $x + J (R)$ is adjacent to every element of $y + J (R)$ .
2.: If $2 x \notin U (R)$ , then $x + J (R)$ is a coclique in $G (R)$ .
3.: If $2 x \in U (R)$ , then $x + J (R)$ is a clique in $G (R)$ .

Proposition 1.

[5] Let R be a finite ring. Then, the following statements hold for the unit graph of R:

1.: If $2 \notin U (R)$ , then the unit graph $G (R)$ is a $| U (R) |$ -regular graph.
2.: If $2 \in U (R)$ , then for every $x \in U (R)$ we have $d e g (x) = | U (R) | - 1$ , and for every $x \in R ∖ U (R)$ we have $d e g (x) = | U (R) |$ .

For a finite simple undirected graph G, the adjacency matrix

A (G)

is defined as the

n \times n

matrix whose

(i, j) t h

entry is 1 if

v_{i} \sim v_{j}

and 0 otherwise. The Laplacian matrix

L (G)

of G is defined by

L (G) : = D (G) - A (G)

, where

D (G) = D i a g (d_{1}, d_{2}, . . ., d_{n})

is the diagonal matrix such that

d_{i}

are degrees of vertices of G. The matrix

L (G)

is a real, symmetric, and positive semidefinite so that its eigenvalues are real and nonnegative. Since the sum of each row of

L (G)

is zero, the smallest eigenvalue of

L (G)

is 0. The largest eigenvalue of

L (G)

is known as the Laplacian spectral radius

λ (G)

of G, and the second smallest eigenvalue of

L (G)

is known as the algebraic connectivity

μ (G)

of G and

μ (G) > 0

if and only if G is connected. A graph G is called Laplacian integral if all the eigenvalues of

L (G)

are integers. More literature about the Laplacian matrix of graphs can be seen in [13,14].

The spectrum of a square matrix C, denoted by

σ (C)

, is the multiset of all the eigenvalues of C. If

η_{1}, η_{2}, . . ., η_{c}

are distinct eigenvalues of B with respective multiplicities

γ_{1}

γ_{2}

, . . ., γ_{c}

, then we shall denote the spectrum of C by

σ (C) = \{\begin{matrix} η_{1} & η_{2} & . . . & η_{c} \\ γ_{1} & γ_{2} & . . . & γ_{c} \end{matrix}\} .

For a graph G, the Laplacian spectrum of G is the spectrum of

L (G)

, we write

σ (L (G))

σ_{L} (G)

. For example,

σ_{L} (K_{n}) = \{\begin{matrix} 0 & n \\ 1 & n - 1 \end{matrix}\} and σ_{L} ({\bar{K}}_{n}) = \{\begin{matrix} 0 \\ n \end{matrix}\} .

(1)

Let G be a graph on k vertices with

V (G) = {v_{1}, v_{2}, . . ., v_{k}}

and let

H_{1}, H_{2}, . . ., H_{k}

be k pairwise disjoint graphs. The G-generalized join graph

G [H_{1}, H_{2}, . . ., H_{k}]

H_{1}, H_{2}, . . ., H_{k}

is the graph formed by replacing each vertex

v_{i}

of G by the graph

H_{i}

and then joining each vertex of

H_{i}

to every vertex of

H_{j}

whenever

v_{i} \sim v_{j}

in G [15]. The following result is useful in the sequel.

Theorem 1.

[16] Let G be a graph on k vertices with

V (G) = {v_{1}, v_{2}, . . ., v_{k}}

and let

H_{1}, H_{2}, . . ., H_{k}

be k pairwise disjoint graphs on

n_{1}, n_{2}, . . ., n_{k}

vertices, respectively. Then, the Laplacian spectrum of

G [H_{1}, H_{2}, . . ., H_{k}]

is given by

σ_{L} (G [H_{1}, H_{2}, . . ., H_{k}]) = (\begin{matrix} ⋃_{j = 1}^{k} (M_{j} + (σ_{L} (H_{j})) ∖ {0}) \end{matrix}) ⋃ σ (L (G)),

(2)

where

M_{j} = \{\begin{matrix} \sum_{v_{i} \sim v_{j}} n_{i} & i f N_{G} (v_{j}) \neq \emptyset; \\ 0 & o t h e r w i s e, \end{matrix}

L (G) = [\begin{matrix} M_{1} & - s_{1, 2} & . . . & - s_{1, k} \\ - s_{1, 2} & M_{2} & . . . & - s_{2, k} \\ . . . & . . . & . . . & . . . \\ - s_{1, k} & - s_{2, k} & . . . & M_{k} \end{matrix}],

and

s_{i, j} = \{\begin{matrix} \sqrt{n_{i} n_{j}} & i f v_{i} \sim v_{j} i n G; \\ 0 & o t h e r w i s e . \end{matrix}

In (2),

σ_{L} (H_{j}) ∖ {0}

means that one copy of the eigenvalue 0 is removed from the multiset

σ_{L} (H_{j})

, and

(M_{j} + (σ_{L} (H_{j})) ∖ {0})

means that

M_{j}

is added to each element of

σ_{L} (H_{j}) ∖ {0}

Let n be a positive integer. Euler’s totient function, denoted by

φ (n)

, is the number of positive integers less than or equal to n that are relatively prime to n. Let

n = p_{1} p_{2} . . . p_{k}

and

p_{i}

be distinct primes for

i = 1, 2, . . ., k

. Note that,

φ (p_{i}) = p_{i} - 1

and

φ (p_{1} p_{2} . . . p_{k}) = φ (p_{1}) φ (p_{2}) . . . φ (p_{k})

, so

φ (n) = \prod_{i = 1}^{k} φ (p_{i})

. Fakieh et al. studied the Laplacian spectrum of the unit graphs associated to the ring

Z_{p_{1} p_{2} . . . p_{k}}

, where

p_{i}

are distinct primes and

i = 1, 2, . . ., k

[4]. This result is the main tool to prove Theorem 4.

Theorem 2.

[4] Let

p_{i} \neq 2

be distinct primes and k be a positive integer,

1 \leq i, j \leq k

. Then:

1.: If $n = p_{1} p_{2} . . . p_{k}$ , then the Laplacian spectrum of $G (Z_{n})$ is

$\begin{matrix} σ_{L} (G (Z_{n})) & = \{\begin{matrix} 0 & [φ (p_{i}) \pm 1] \prod_{i \neq j} φ (p_{j}) & [φ (p_{i}) φ (p_{j}) \pm 1] \prod_{h \neq i, j} φ (p_{h}) & . . . \\ 1 & \frac{φ (p_{i})}{2} & \frac{φ (p_{i}) φ (p_{j})}{2} & . . . \end{matrix} \\ \begin{matrix} [φ (p_{1}) φ (p_{2}) . . . φ (p_{k - 1}) \pm 1] φ (p_{k}) & [\prod_{1 \leq i \leq k} φ (p_{i})] \pm 1 \\ \frac{\prod_{1 \leq i \leq k - 1} φ (p_{i})}{2} & \frac{\prod_{1 \leq i \leq k} φ (p_{i})}{2} \end{matrix}\} . \end{matrix}$
2.: If $n = 2 p_{1} p_{2} . . . p_{k}$ , then the Laplacian spectrum of $G (Z_{n})$ is

$\begin{matrix} σ_{L} (G (Z_{n})) & = \{\begin{matrix} 0 & [φ (p_{i}) \pm 1] \prod_{i \neq j} φ (p_{j}) & [φ (p_{i}) φ (p_{j}) \pm 1] \prod_{h \neq i, j} φ (p_{h}) & . . . \\ 1 & φ (p_{i}) & φ (p_{i}) φ (p_{j}) & . . . \end{matrix} \\ \begin{matrix} [φ (p_{1}) φ (p_{2}) . . . φ (p_{k - 1}) \pm 1] φ (p_{k}) & [\prod_{1 \leq i \leq k} φ (p_{i})] \pm 1 & 2 \prod_{1 \leq i \leq k} φ (p_{i}) \\ \prod_{1 \leq i \leq k - 1} φ (p_{i}) & \prod_{1 \leq i \leq k} φ (p_{i}) & 1 \end{matrix}\} . \end{matrix}$

The vertex connectivity

κ (G)

of a graph G is the minimum number of vertices whose removal from G leaves a disconnected or trivial graph. A family of two or more paths in a graph G is said to be internally disjoint if no vertex of G is an internal vertex of more than one path in the family. There is a strong result for the vertex connectivity

κ (G)

by Menger’s theorem. Menger’s theorem says that the maximum number of internally disjoint

u v

-paths in G is equal to the minimum number of vertices whose deletion destroys all

u v

-paths, where u and v are nonadjacent in G [17]. So that

\begin{matrix} κ (G) & = min {ρ (u, v) : ρ (u, v) is the maximum number of internally disjoint u v - paths in G, \\ where u ≁ v} . \end{matrix}

This paper uses Menger’s theorem to examine the vertex connectivity of

G (Z_{n})

3. $G (Z_{n})$ as a Generalized Join Graph

For a positive integer n,

Z_{n}

denotes the ring of integers modulo n. The elements of the ring

Z_{n}

are referred to as

0, 1, 2,

and

n - 1

. A nonzero element

x \in Z_{n}

is a unit in

Z_{n}

if x is relatively prime with n,

(x, n) = 1

. Through this section, we use

p_{i}

as a prime number. Also, an integer n can be written in the form

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

p_{1}, p_{2}, . . ., p_{k}

are distinct primes and

k, r_{1}, r_{2}, . . ., r_{k}

are positive integers such that at least one of the

r_{i}

must be greater than 1. In this section, we prove that

G (Z_{n})

is a generalized join graph of some complete graphs and null graphs. To this end, first, we study the structure of

G (Z_{n})

. Denote the maximal ideal of

Z_{n}

m_{i}

m_{i}

is an ideal generated by the prime divisors

p_{i}

of n, that is,

m_{i} = 〈p_{i}〉

. So,

J (Z_{n}) = \cap m_{i} = 〈\prod_{i = 1}^{k} p_{i}〉 \neq {0}

. Put

p = \prod_{i = 1}^{k} p_{i}

, then

Z_{n} / J (Z_{n}) = {i + J (Z_{n}) : i \in {0, 1, . . ., p - 1}} and Z_{n} / J (Z_{n}) ≅ Z_{p} .

Let S be the set of distinct representatives of

Z_{n} / J (Z_{n})

. For

i \in S

, we denote

B_{i} = i + J (Z_{n}) .

Note that the sets

B_{0}, B_{1}, . . ., B_{p - 1}

form a partition of the vertex set of

G (Z_{n})

. Thus,

V (G (Z_{n})) = ⋃_{i = 0}^{p - 1} B_{i} .

(3)

Let

B_{0} = J (Z_{n})

. Then, for

j \in S

| B_{j} | = | B_{0} | = | J (Z_{n}) | = p_{1}^{r_{1} - 1} p_{2}^{r_{2} - 1} . . . p_{k}^{r_{k} - 1} .

The following result describes the adjacency criterion of vertices in

G (Z_{n})

, where

V (G (Z_{n}))

is described in Equation (3).

Lemma 2.

For

i, j \in S

, every vertex of

B_{i}

is adjacent to every vertex of

B_{j}

G (Z_{n})

if and only if

(i + j, n) = 1

Proof.

(\Rightarrow)

Clearly.

(\Leftarrow)

Suppose that

(i + j, n) = 1

. Let

a \in B_{i}

and

b \in B_{j}

, which can be written as

a = i + a_{i}

and

b = j + b_{j}

, where

a_{i}, b_{j} \in J (Z_{n})

. Now,

a + b = (i + j) + (a_{i} + b_{j})

. Here

i + j

is a unit by assumption, and

a_{i} + b_{j}

is nilpotent. So,

a + b

is a unit, and hence every vertex of

B_{i}

is adjacent to every vertex of

B_{j}

G (Z_{n})

. □

By using Lemmas 1 [(2),(3)] and 2, the following is evident.

Corollary 1.

The following statements hold:

1.: For $i \in S$ , the induced subgraph $G (B_{i})$ of $G (Z_{n})$ on the vertex set $B_{i}$ is either the complete graph $K_{| J (Z_{n}) |}$ or its complement graph ${\bar{K}}_{| J (Z_{n}) |}$ . Indeed, $G (B_{i})$ is $K_{| J (Z_{n}) |}$ if and only if $2 i \in U (Z_{n})$ .
2.: For $i, j \in S$ with $i \neq j$ , a vertex of $G (B_{i})$ is adjacent to either all or none of the vertices of $G (B_{j})$ in $G (Z_{n})$ .

The above corollary implies that the partition

B_{0} \cup B_{1} \cup . . . \cup B_{p - 1}

of the vertex set

V (G (Z_{n}))

G (Z_{n})

is an equitable partition in such a way that every vertex of the

B_{i}

has equal number of neighbors in

B_{j}

for all

i, j \in S

We define

G_{n}

by the simple graph whose vertices are the distinct representatives of

Z_{n} / J (Z_{n})

, that is, the set of vertices is S, and in which two distinct vertices i and j are adjacent if and only if

(i + j, n) = 1

. The graph

G_{n}

will play an important role in the rest of the paper.

Lemma 3.

G (Z_{p}) ≅ G_{n}

Proof.

We define a map

ϕ : V (G (Z_{p})) \to V (G_{n})

such that

ϕ (B_{i}) = i

. Clearly,

ϕ

is well-defined and bijection. From Lemma 2, the adjacency relationships are preserved by

ϕ

. Hence, the result follows. □

The following lemma states that

G (Z_{n})

can be expressed as a generalized join of certain complete graphs and null graphs.

Lemma 4.

Let

G (B_{i})

be the induced subgraph of

G (Z_{n})

on the vertex set

B_{i}

for

0 \leq i \leq p - 1

. Then,

G (Z_{n}) = G_{n} [G (B_{0}), G (B_{1}), \dots, G (B_{p - 1})] .

Proof.

Replace the vertex i of

G_{n}

G (B_{i})

for

0 \leq i \leq p - 1

. Thus, the result follows from Lemma 2 and Corollary 1. □

Example 1.

The unit graph

G (Z_{25})

is shown in Figure 1. By Lemma 4, we have

G (Z_{25}) = G_{25} [G (B_{0}), G (B_{1}), G (B_{2}), G (B_{3}), G (B_{4})],

where

G_{25}

is shown in Figure 2,

G (B_{0}) = {\bar{K}}_{5}

, and

G (B_{i}) = K_{5}

for

1 \leq i \leq 4

. In Figure 1, the lines between two squares mean that each vertex in one square is adjacent to every vertex in the other square.

4. Laplacian Spectrum of $G (Z_{n})$

In this section, we investigate the Laplacian spectrum of

G (Z_{n})

for

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

2 \neq p_{1} < p_{2} < . . . < p_{k}

are primes. For

0 \leq i \leq p - 1

, we give the weight

| J (Z_{n}) | = t

to the vertex i of the graph

G_{n}

. Define the integer

M_{j} = t b_{j}, w h e r e b_{j} = d e g (j) i n G_{n}

for

0 \leq j \leq p - 1

. The

p \times p

vertex weighted Laplacian matrix

L (G_{n})

G_{n}

defined in Theorem 1 is given by

L (G_{n}) = [\begin{matrix} M_{0} & - h_{0, 1} & . . . & - h_{0, p - 1} \\ - h_{0, 1} & M_{1} & . . . & - h_{1, p - 1} \\ . . . & . . . & . . . & . . . \\ - h_{0, p - 1} & - h_{1, p - 1} & . . . & M_{p - 1} \end{matrix}],

(4)

where

h_{i, j} = \{\begin{matrix} t & i f i \sim j i n G_{n}; \\ 0 & o t h e r w i s e \end{matrix}

for

0 \leq i \neq j \leq p - 1

The following remark is an immediate result of Proposition 1.

Remark 1.

The following statements hold:

1.: If $2 \notin U (Z_{p})$ , then $M_{j} = t φ (p)$ .
2.: If $2 \in U (Z_{p})$ , then

$M_{j} = \{\begin{matrix} t (φ (p) - 1) & i f j \in U (Z_{p}); \\ t φ (p) & i f j \notin U (Z_{p}) . \end{matrix}$

Lemma 5.

L (G_{n}) = t (L (G_{n}))

Proof.

The proof is direct from definition of

L (G_{n})

in Equation (4). □

The following theorem describes the Laplacian spectrum of

G (Z_{n})

Theorem 3.

Let

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

p_{1} < p_{2} < . . . < p_{k}

are primes,

r_{i}

and k are positive integers, and

1 \leq i \leq k

. The Laplacian spectrum of

G (Z_{n})

is given by

σ_{L} (G (Z_{n})) = (\begin{matrix} ⋃_{j = 0}^{p - 1} (M_{j} + (σ_{L} (G (B_{j})) ∖ {0})) \end{matrix}) ⋃ t σ_{L} (G_{n}),

where

j \in S

and

M_{j} + (σ_{L} (G (B_{j}) ∖ {0})

means that

M_{j}

is added to each element of the multiset

σ_{L} (G (B_{j}) ∖ {0}

Proof.

By Lemma 4, we have

G (Z_{n}) = G_{n} [G (B_{0}), G (B_{1}), \dots, G (B_{p - 1})] .

Consequently, the result can be obtained by using Theorem 1 and Lemma 5. □

By Corollary 1,

G (B_{i})

is either

K_{t}

{\bar{K}}_{t}

for

1 \leq i \leq k

. By Theorem 3, out of the n number of Laplacian eigenvalues of

G (Z_{n})

n - p

of them are known to be nonzero integer values. The remaining

p

Laplacian eigenvalues of

G (Z_{n})

will come from the Laplacian eigenvalues of

G (Z_{p})

Corollary 2.

Let

2 \neq p_{1} < p_{2} < . . . < p_{k}

be distinct primes and

r,

r_{i}

s_{i}

, k be positive integers, where

1 \leq i \leq k

. Then, the Laplacian spectrum of

G (Z_{n})

is given by

1.: If $n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}$ , then

$\begin{matrix} σ_{L} (G (Z_{n})) & = (\underset{φ (p) - times}{\underset{︸}{t (φ (p) - 1) + (σ_{L} (K_{t}) ∖ {0})}}) ⋃ (\underset{[p - φ (p)] - times}{\underset{︸}{t (φ (p) + (σ_{L} ({\bar{K}}_{t}) ∖ {0})}}) ⋃ t σ_{L} (G_{n}) . \end{matrix}$
2.: If $n = 2^{r} p_{1}^{s_{1}} p_{2}^{s_{2}} . . . p_{k}^{s_{k}}$ , then

$\begin{matrix} σ_{L} (G (Z_{n})) & = (\underset{2 p - times}{\underset{︸}{t (φ (p) + (σ_{L} ({\bar{K}}_{t}) ∖ {0})}}) ⋃ t σ_{L} (G_{n}) . \end{matrix}$

Proof.

By the above argument and Remark 1, the result holds. □

The following result gives the Laplacian spectrum of

G (Z_{n})

for

n = p^{r} q^{s}

, where

p < q

are primes and r, s are positive integers.

Theorem 4.

Let

2 \neq p < q

be primes and

r, s

be positive integers. Then:

1.: If $n = p^{r} q^{s}$ , then the Laplacian spectrum of $G (Z_{n})$ consists of

$\begin{matrix} \{\begin{matrix} 0 & p^{r - 1} q^{s - 1} [φ (p) - 1] φ (q) & p^{r - 1} q^{s - 1} [φ (q) - 1] φ (p) & p^{r - 1} q^{s - 1} [φ (p) φ (q) - 1] \\ 1 & \frac{φ (p)}{2} & \frac{φ (q)}{2} & \frac{φ (p) φ (q)}{2} \end{matrix} \\ \begin{matrix} p^{r - 1} q^{s - 1} [φ (p) φ (q) + 1] & p^{r - 1} q^{s} φ (p) & p^{r} q^{s - 1} φ (q) & p^{r - 1} q^{s - 1} φ (p) φ (q) \\ \frac{φ (p) φ (q)}{2} & \frac{φ (q)}{2} & \frac{φ (p)}{2} & p^{r} q^{s} - p q \end{matrix}\} . \end{matrix}$

(5)
2.: If $n = 2^{r} q^{s}$ , then the Laplacian spectrum of $G (Z_{n})$ is

$\begin{matrix} σ_{L} (G (Z_{n})) = \{\begin{matrix} 0 & 2^{r - 1} p^{s - 1} [φ (p) - 1] & 2^{r - 1} p^{s} & 2^{r} p^{s - 1} φ (p) & 2^{r - 1} p^{s - 1} φ (p) \\ 1 & φ (p) & φ (p) & 1 & 2^{r} p^{s} - 2 p \end{matrix}\} . \end{matrix}$

Proof.

1. Let

n = p^{r} q^{s}

, where

2 < p < q

are primes and r, s are positive integers. So, the Jacobson radical of

Z_{n}

〈p q〉

and the set of distinct representatives of

Z_{n} / 〈p q〉

{0, 1, . . ., p q - 1}

. Thus,

G (Z_{p^{r} q^{s}}) = G_{p^{r} q^{s}} [\underset{φ (p q) - times}{\underset{︸}{K_{p^{r - 1} q^{s - 1}}, \dots, K_{p^{r - 1} q^{s - 1}}}}, \underset{[p q - φ (p q)] - times}{\underset{︸}{{\bar{K}}_{p^{r - 1} q^{s - 1}}, \dots, {\bar{K}}_{p^{r - 1} q^{s - 1}}}}] .

By Corollary 2, the Laplacian spectrum of

G (Z_{n})

is given by

\begin{matrix} σ_{L} (G (Z_{n})) & = (\underset{φ (p q) - times}{\underset{︸}{t (φ (p q) - 1) + (σ_{L} (K_{p^{r - 1} q^{s - 1}}) ∖ {0})}}) ⋃ (\underset{[p q - φ (p q)] - times}{\underset{︸}{t (φ (p q) + (σ_{L} ({\bar{K}}_{t}) ∖ {0})}}) \\ ⋃ p^{r - 1} q^{s - 1} σ_{L} (G_{p^{r} q^{s}}) . \end{matrix}

By Eq (1), the Laplacian spectrum of

K_{p^{r - 1} q^{s - 1}}

and

{\bar{K}}_{p^{r - 1} q^{s - 1}}

are

σ_{L} (K_{p^{r - 1} q^{s - 1}}) = \{\begin{matrix} 0 & p^{r - 1} q^{s - 1} \\ 1 & p^{r - 1} q^{s - 1} - 1 \end{matrix}\} and σ_{L} ({\bar{K}}_{p^{r - 1} q^{s - 1}}) = \{\begin{matrix} 0 \\ p^{r - 1} q^{s - 1} \end{matrix}\} .

Then,

\begin{matrix} σ_{L} (G (Z_{n})) & = \{\begin{matrix} p^{r - 1} q^{s - 1} [φ (p q) - 1] + p^{r - 1} q^{s - 1} \\ φ (p q) (p^{r - 1} q^{s - 1} - 1) \end{matrix}\} ⋃ \{\begin{matrix} p^{r - 1} q^{s - 1} φ (p q) \\ p q - φ (p q) (p^{r - 1} q^{s - 1} - 1) \end{matrix}\} \\ ⋃ \{p^{r - 1} q^{s - 1} σ_{L} (G_{p^{r} q^{s}})\} . \end{matrix}

By using Lemma 3,

G_{p^{r} q^{s}}

is isomorphic to

G (Z_{p q})

, and hence

σ_{L} (G_{p^{r} q^{s}}) = σ_{L} (G (Z_{p q}))

. So, by Theorem 2, we have

\begin{matrix} σ_{L} (G_{p^{r} q^{s}}) & = \{\begin{matrix} 0 & [φ (p) - 1] φ (q) & [φ (q) - 1] φ (p) & φ (p) φ (q) - 1 & φ (p) φ (q) + 1 \\ 1 & \frac{φ (p)}{2} & \frac{φ (q)}{2} & \frac{φ (p) φ (q)}{2} & \frac{φ (p) φ (q)}{2} \end{matrix} \\ \begin{matrix} φ (p) [φ (q) + 1] & φ (q) [φ (p) + 1] \\ \frac{φ (q)}{2} & \frac{φ (p)}{2} \end{matrix}\} . \end{matrix}

So,

\begin{matrix} σ_{L} (G (Z_{p^{r} q^{s}})) & = \{\begin{matrix} p^{r - 1} q^{s - 1} φ (p q) \\ p^{r} q^{s} - p q \end{matrix}\} ⋃ \{\begin{matrix} 0 & p^{r - 1} q^{s - 1} [φ (p) - 1] φ (q) & p^{r - 1} q^{s - 1} [φ (q) - 1] φ (p) \\ 1 & \frac{φ (p)}{2} & \frac{φ (q)}{2} \end{matrix} \\ \begin{matrix} p^{r - 1} q^{s - 1} [φ (p) φ (q) - 1] & p^{r - 1} q^{s - 1} [φ (p) φ (q) + 1] & p^{r - 1} q^{s} φ (p) & p^{r} q^{s - 1} φ (q) \\ \frac{φ (p) φ (q)}{2} & \frac{φ (p) φ (q)}{2} & \frac{φ (q)}{2} & \frac{φ (p)}{2} \end{matrix}\} . \end{matrix}

Hence, the Laplacian spectrum of

G (Z_{p^{r} q^{s}})

is as in Eq (5).

2. Let

n = 2^{r} p^{s}

, where

p \neq 2

is a prime and

r,

s are positive integers. Note that,

S = {0, 1, . . ., 2 p - 1}

is the vertex set of the graph

G_{2^{r} p^{s}}

. Thus,

G (Z_{2^{r} p^{s}}) = G_{2^{r} p^{s}} [\underset{2 p - times}{\underset{︸}{{\bar{K}}_{2^{r - 1} p^{s - 1}}, {\bar{K}}_{2^{r - 1} p^{s - 1}}, \dots, {\bar{K}}_{2^{r - 1} p^{s - 1}}}}] .

By Corollary 2, the Laplacian spectrum of

G (Z_{n})

is given by

\begin{matrix} σ_{L} (G (Z_{2^{r} p^{s}})) & = (\underset{2 p - times}{\underset{︸}{2^{r - 1} p^{s - 1} (φ (p) + (σ_{L} ({\bar{K}}_{2^{r - 1} p^{s - 1}}) ∖ {0})}}) ⋃ 2^{r - 1} p^{s - 1} σ_{L} (G_{2^{r} p^{s}}) \\ = \{\begin{matrix} 2^{r - 1} p^{s - 1} φ (p) \\ 2 p (2^{r - 1} p^{s - 1} - 1) \end{matrix}\} ⋃ \{2^{r - 1} p^{s - 1} σ_{L} (G_{2^{r} p^{s}})\} . \end{matrix}

By using Lemma 3,

G_{2^{r} p^{s}}

is isomorphic to

G (Z_{2 p})

, and hence

σ_{L} (G_{2^{r} p^{s}}) = σ_{L} (G (Z_{2 p}))

. So, by Theorem 2, we have

σ_{L} (G_{2^{r} p^{s}}) = \{\begin{matrix} 0 & φ (p) - 1 & φ (p) + 1 & 2 φ (p) \\ 1 & φ (p) & φ (p) & 1 \end{matrix}\} .

Hence, the Laplacian spectrum of

G (Z_{n})

\begin{matrix} σ_{L} (G (Z_{n})) = \{\begin{matrix} 0 & 2^{r - 1} p^{s - 1} [φ (p) - 1] & 2^{r - 1} p^{s} & 2^{r} p^{s - 1} φ (p) & 2^{r - 1} p^{s - 1} φ (p) \\ 1 & φ (p) & φ (p) & 1 & 2^{r} p^{s} - 2 p \end{matrix}\} . \end{matrix}

□

Now, we find the Laplacian spectrum of

G (Z_{n})

for

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

p_{1} < p_{2} < . . . < p_{k}

are primes,

r_{i}, k

are positive integers, and

1 \leq i \leq k

. The following theorem can be obtained by arguments similar to those used in the proof of Theorem 4, and therefore the proof is omitted.

Theorem 5.

Let

2 \neq p_{1} < p_{2} < . . . < p_{k}

be primes and

r, r_{i}, s_{i}, k

be positive integers,

1 \leq i \leq k

. Then:

1.: If $n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}$ , then the Laplacian spectrum of $G (Z_{n})$ consists of

$\begin{matrix} \{\begin{matrix} 0 & [φ (p_{i}) \pm 1] \prod_{1 \leq i \leq k} p_{i}^{(r_{i} - 1)} \prod_{i \neq j} φ (p_{j}) & [φ (p_{i}) φ (p_{j}) \pm 1] \prod_{1 \leq i \leq k} p_{i}^{(r_{i} - 1)} \prod_{h \neq i, j} φ (p_{h}) & . . . \\ 1 & \frac{φ (p_{i})}{2} & \frac{φ (p_{i}) φ (p_{j})}{2} & . . . \end{matrix} \\ \begin{matrix} [φ (p_{1}) φ (p_{2}) . . . φ (p_{k - 1}) \pm 1] φ (p_{k}) \prod_{1 \leq i \leq k} p_{i}^{(r_{i} - 1)} & [[\prod_{1 \leq i \leq k} φ (p_{i})] \pm 1] \prod_{1 \leq i \leq k} p_{i}^{(r_{i} - 1)} \\ \frac{\prod_{1 \leq i \leq k - 1} φ (p_{i})}{2} & \frac{\prod_{1 \leq i \leq k} φ (p_{i})}{2} \end{matrix} \\ \begin{matrix} \prod_{1 \leq i \leq k} p_{i}^{(r_{i} - 1)} φ (p_{i}) \\ n - p_{1} p_{2} . . . p_{k} \end{matrix}\} . \end{matrix}$
2.: If $n = 2^{r} p_{1}^{s_{1}} p_{2}^{s_{2}} . . . p_{k}^{s_{k}}$ , then the Laplacian spectrum of $G (Z_{n})$ consists of

\begin{matrix} \{\begin{matrix} 0 & 2^{r - 1} [φ (p_{i}) \pm 1] \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} \prod_{i \neq j} φ (p_{j}) & 2^{r - 1} [φ (p_{i}) φ (p_{j}) \pm 1] \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} \prod_{h \neq i, j} φ (p_{h}) & . . . \\ 1 & φ (p_{i}) & φ (p_{i}) φ (p_{j}) & . . . \end{matrix} \\ \begin{matrix} 2^{r - 1} [φ (p_{1}) φ (p_{2}) . . . φ (p_{k - 1}) \pm 1] φ (p_{k}) \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} & 2^{r - 1} [[\prod_{1 \leq i \leq k} φ (p_{i})] \pm 1] \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} \\ \prod_{1 \leq i \leq k - 1} φ (p_{i}) & \prod_{1 \leq i \leq k} φ (p_{i}) \end{matrix} \\ \begin{matrix} 2^{r} \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} φ (p_{i}) & 2^{(r - 1)} \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} φ (p_{i}) \\ 1 & n - 2 p_{1} p_{2} . . . p_{k} \end{matrix}\} . \end{matrix}

As an immediate consequence of Theorem 5, we have the following results.

Corollary 3.

G (Z_{n}

) is Laplacian integral for all n.

Corollary 4.

The Laplacian spectral radius of

G (Z_{n})

λ (G (Z_{n})) = \{\begin{matrix} [φ (p_{1}) + 1] \prod_{1 \leq i \leq k} p_{i}^{(r_{i} - 1)} \prod_{j \neq 1} φ (p_{j}) & if n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}; \\ 2^{r} \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} φ (p_{i}) & if n = 2^{r} p_{1}^{s_{1}} p_{2}^{s_{2}} . . . p_{k}^{s_{k}}, \end{matrix}

where

2 \neq p_{1} < p_{2} < . . . < p_{k}

are primes,

r, r_{i}, s_{i}, k

are positive integers, and

1 \leq i \leq k

Corollary 5.

The algebraic connectivity of

G (Z_{n})

μ (G (Z_{n})) = \{\begin{matrix} [φ (p_{1}) - 1] \prod_{1 \leq i \leq k} p_{i}^{(r_{i} - 1)} \prod_{j \neq 1} φ (p_{j}) & if n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}; \\ 2^{r - 1} [φ (p_{1}) - 1] \prod_{1 \leq i \leq k} p_{i}^{(s_{i} - 1)} \prod_{j \neq 1} φ (p_{j}) & if n = 2^{r} p_{1}^{s_{1}} p_{2}^{s_{2}} . . . p_{k}^{s_{k}}, \end{matrix}

where

2 \neq p_{1} < p_{2} < . . . < p_{k}

are primes,

r, r_{i}, s_{i}, k

are positive integers, and

1 \leq i \leq k

5. Vertex Connectivity of $G (Z_{n})$

In this section, we obtain the vertex connectivity of

G (Z_{n})

when

n = p q

and

n = p^{r} q^{s}

, where

2 \neq p < q

are primes and r and s are positive integers. To achieve this goal, we calculate the number of internally disjoint paths between any two nonadjacent vertices in

G (Z_{p q})

, which allow us to be ready to explore the vertex connectivity of

G (Z_{p q})

by using Menger’s theorem. We end this section by presenting conjectures about the vertex connectivity of

G (Z_{n})

when

n = p_{1} p_{2} . . . p_{k}

and

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

p_{i}

are distinct primes,

r_{i}

are positive integers, and

1 \leq i \leq k

5.1. Structure of $G (Z_{p q})$

The following result will be used in the sequel.

Lemma 6.

[18] Let

n = p q

, where p and q are distinct odd primes. Then the following statements hold:

1.: Let $i \in {1, 2, . . ., p - 1}$ . The induced subgraph $G (i + 〈p〉)$ of $G (Z_{n})$ is isomorphic to $K_{1} \lor C P (q - 1)$ 1. If $i = 0$ , then $G (i + 〈p〉)$ is ${\bar{K}}_{q}$ .
2.: Let $i, j \in {0, 1, . . ., p - 1}$ and $i \neq j$ . If $(i + j, p) = 1$ , then every vertex of $G (i + 〈p〉)$ is adjacent to $(q - 1)$ vertices of $G (j + 〈p〉)$ .
3.: Let $i, j \in {1, 2, . . ., p - 1}$ and $i \neq j$ . If $(i + j, p) \neq 1$ , then every vertex of $G (i + 〈p〉)$ is nonadjacent to any vertex of $G (j + 〈p〉)$ .

Now, we study the structure of

G (Z_{p q})

, where

2 \neq p < q

are primes, analogously to Section 3. In this case, we choose the maximal ideal

〈p〉

. Let

S^{'}

be the set of distinct representatives of

Z_{p q} / 〈p〉

. For

i \in S^{'}

, we denote

B_{i}^{'} = i + 〈p〉 .

Note that the sets

B_{0}^{'}, B_{1}^{'}, . . ., B_{p - 1}^{'}

form a partition of the vertex set of

G (Z_{p q})

. Thus,

V (G (Z_{n})) = ⋃_{i = 0}^{p - 1} B_{i}^{'} .

This implies that any two vertices x and y that belong to the above union are adjacent if and only if

(x + y, p q) = 1

. Let

B_{0}^{'} = 〈p〉

. Then,

| B_{i}^{'} | = | B_{0}^{'} | = | 〈p〉 | = q

for

i \in S^{'}

Note that, Lemma 6 implies that the partition

\cup_{i = 0}^{p - 1} G (B_{i}^{'})

V (G (Z_{p q}))

is an almost equitable partition in such a way that every vertex of

G (B_{i}^{'})

has an equal number of neighbors in

G (B_{j}^{'})

where

i \neq j

and

i, j \in S^{'}

. Also,

G (B_{0}^{'})

is isomorphic to

{\bar{K}}_{q}

and

G (B_{i}^{'})

is isomorphic to

K_{1} \lor C P (q - 1)

, where

i \in S^{'} - {0}

Let

G_{p q}^{'}

be defined as the simple graph whose vertices are

G (B_{i}^{'})

, where

i \in S^{'}

, so that

| V (G_{p q}^{'}) | = p

. Two distinct vertices

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

are adjacent if and only if

(i + j, p) = 1

, which is equivalent to each vertex in

G (B_{i}^{'})

being adjacent to

(q - 1)

vertices in

G (B_{j}^{'})

Example 2.

The vertex set of

G (Z_{15})

, denoted as

V (G_{15}^{'})

, can be expressed as the union

V (G_{15}^{'}) = G (B_{0}^{'}) \cup G (B_{1}^{'}) \cup G (B_{2}^{'})

, where

G (B_{0}^{'}) = G (0 + 〈3〉)

G (B_{1}^{'}) = G (1 + 〈3〉)

, and

G (B_{2}^{'}) = G (2 + 〈3〉)

. From Figure 3 bellow, we observe the following:

1.: $G (B_{1}^{'})$ is nonadjacent to $G (B_{2}^{'})$ since $(1 + 2, 3) \neq 1$ .
2.: Each vertex in $G (B_{1}^{'})$ and $G (B_{2}^{'})$ is adjacent to 4 vertices in $G (B_{0}^{'})$ .
3.: $G (B_{0}^{'})$ is isomorphic to ${\bar{K}}_{5}$ and $G (B_{1}^{'})$ and $G (B_{2}^{'})$ are isomorphic to $K_{1} \lor C P (4)$ . The red vertices 0, 5, and 10 represent $K_{1}$ in $G (B_{0}^{'})$ , $G (B_{2}^{'})$ , and $G (B_{1}^{'})$ , respectively. Note that these vertices are multiples of 5.

The following two results determine the neighbors and the number of common neighbors of the vertices in

G_{p q}^{'}

, which help us to calculate the number of internally disjoint paths between any two nonadjacent vertices in

\cup_{i = 0}^{p - 1} G (B_{i}^{'})

Lemma 7.

Let

i \in S^{'}

. If

i \neq 0

, then there are

(p - 2)

neighbors of

G (B_{i}^{'})

G_{p q}^{'}

. On the other hand,

G (B_{0}^{'})

has

(p - 1)

neighbors in

G_{p q}^{'}

Proof.

Let

i \neq 0

. For

j \neq i

(i + j, p) = 1

for all

j \in S^{'}

except

j = p - i

. Consequently,

G (B_{i}^{'})

is adjacent to

G (B_{j}^{'})

G_{p q}^{'}

when

j \neq i, p - i

. Then, there are

(p - 2)

neighbors of

G (B_{i})

G_{p q}^{'}

. If

i = 0

, then

(0 + j, p) = 1

for all

j \neq i

. Thus,

G (B_{0}^{'})

is adjacent to all vertices in

G_{p q}^{'}

. Therefore, there are

(p - 1)

neighbors of

G (B_{0}^{'})

G_{p q}^{'}

. □

Lemma 8.

i, j \in S^{'}

, then the following statements hold:

1.: If $G (B_{i}^{'})$ and $G (B_{j}^{'})$ are nonadjacent in $G_{p q}^{'}$ , then the number of common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ is $(p - 2)$ .
1.: If $G (B_{i}^{'})$ and $G (B_{j}^{'})$ are adjacent in $G_{p q}^{'}$ , where $i, j \neq 0$ , then the number of common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ is $(p - 4)$ .
3.: The number of common neighbors between $G (B_{0}^{'})$ and $G (B_{i}^{'})$ in $G_{p q}^{'}$ is $(p - 3)$ .

Proof.

Let

i, j \in S^{'}

Let $G (B_{i}^{'})$ and $G (B_{j}^{'})$ be nonadjacent in $G_{p q}^{'}$ . By Lemma 7, $j = p - i$ and $i = p - j$ . Hence, $G (B_{i}^{'})$ is adjacent to all vertices in $G_{p q}^{'}$ except $G (B_{p - i}^{'})$ . Similarly, $G (B_{j}^{'})$ is adjacent to all vertices in $G_{p q}^{'}$ except $G (B_{p - j}^{'})$ . So, $G (B_{i}^{'})$ and $G (B_{j}^{'})$ are adjacent to all vertices in $G_{p q}^{'}$ except $G (B_{j}^{'})$ and $G (B_{i}^{'})$ . Therefore, there are $(p - 2)$ common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ .
Let $G (B_{i}^{'})$ and $G (B_{j}^{'})$ be adjacent in $G_{p q}^{'}$ , where $i, j \neq 0$ . According to Lemma 7, $G (B_{i}^{'})$ and $G (B_{j}^{'})$ are nonadjacent to $G (B_{p - i}^{'})$ and $G (B_{p - j}^{'})$ in $G_{p q}^{'}$ , respectively. So, $G (B_{i}^{'})$ and $G (B_{j}^{'})$ are adjacent to all vertices in $G_{p q}^{'}$ except $G (B_{p - i}^{'})$ and $G (B_{p - j}^{'})$ , respectively. Then, the set of common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ is

$V (G_{p q}^{'}) - {G (B_{i}^{'}), G (B_{j}^{'}), G (B_{p - i}^{'}), G (B_{p - j}^{'})} .$

Thus, there are $(p - 4)$ common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ .
By Lemma 7, $G (B_{0}^{'})$ is adjacent to all vertices in $G_{p q}^{'}$ . Also, $G (B_{i}^{'})$ is adjacent to all vertices in $G_{p q}^{'}$ except $G (B_{p - i}^{'})$ . So, the set of common neighbors between $G (B_{0}^{'})$ and $G (B_{i}^{'})$ in $G_{p q}^{'}$ is

$V (G_{p q}^{'}) - {G (B_{0}^{'}), G (B_{i}^{'}), G (B_{p - i}^{'})} .$

Then, there are $(p - 3)$ common neighbors between $G (B_{0}^{'})$ and $G (B_{i}^{'})$ in $G_{p q}^{'}$ . □

The following result determines the number of common neighbors between nonadjacent vertices

x, y \in G (B_{i}^{'})

through

G (B_{z}^{'})

, where

G (B_{z}^{'})

is a neighbor of

G (B_{i}^{'})

G_{p q}^{'}

Lemma 9.

Let

x, y \in G (B_{i}^{'})

be nonadjacent and

G (B_{z}^{'})

be a neighbor of

G (B_{i}^{'})

G_{p q}^{'}

. Then, x and y have

(q - 2)

common neighbors in

G (B_{z}^{'})

Proof.

By Part 2 of Lemma 6, both x and y are adjacent to

(q - 1)

vertices in

G (B_{z}^{'})

. Suppose that x and y have the same neighbors in

G (B_{z}^{'})

. Then, x and y are adjacent to all vertices in

G (B_{z}^{'})

except

z^{'}

. This implies that

z^{'}

is adjacent to

(q - 2)

vertices in

G (B_{i}^{'})

, a contradiction with Part 2 of Lemma 6. Then, x and y are adjacent to all vertices in

G (B_{z}^{'})

except

a_{z}

and

b_{z}

, respectively. So, the number of common neighbors between x and y in

G (B_{z}^{'})

(q - 2)

. □

The following result determines the number of common neighbors between nonadjacent vertices

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

through

G (B_{z}^{'})

, where

G (B_{z}^{'})

is a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

Lemma 10.

Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

be nonadjacent and

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. Then:

1.: If x and y have the same neighbors in $G (B_{z}^{'})$ , then the number of common neighbors in $G (B_{z}^{'})$ between x and y is $(q - 1)$ .
2.: If x and y do not have the same neighbors in $G (B_{z}^{'})$ , then the number of common neighbors in $G (B_{z}^{'})$ between x and y is $(q - 2)$ .

Proof.

Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

be nonadjacent and

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

1.: The proof is direct from Part 2 of Lemma 6.
2.: Let x and y do not have the same neighbors in $G (B_{z}^{'})$ . By Part 2 of Lemma 6, both x and y have $(q - 1)$ neighbors in $G (B_{z}^{'})$ . That is, x and y are adjacent to all vertices in $G (B_{z}^{'})$ except $a_{z}$ and $b_{z}$ , respectively. So, the number of the common neighbors in $G (B_{z}^{'})$ between x and y is $(q - 2)$ . □

From now to the rest of this section, we denote x, which is a multiple of q in

G (B_{i}^{'})

, by

x^{*}

(see Example 2). The following proposition characterizes the nonadjacent vertices of

G (Z_{p q})

for which the relation in Part 1 of Lemma 10 holds when

G (B_{i}^{'})

is adjacent to

G (B^{'} j)

G^{'} p q

Proposition 2.

Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

be nonadjacent,

G (B_{i}^{'})

be adjacent to

G (B_{j}^{'})

G_{p q}^{'}

, and

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. Then, x and y have the same neighbors in

G (B_{z}^{'})

if and only if

x = x^{*}

and

y = y^{*}

Proof.

(\Rightarrow)

Suppose that x and y have the same neighbors in

G (B_{z}^{'})

. Then, x and y are adjacent to all vertices in

G (B_{z}^{'})

except

z^{'}

by Part 2 of Lemma 6. We assume that

x \neq x^{*}

and

y \neq y^{*}

. Suppose that

x = i + a^{'} p

y = j + b^{'} p

, and

z^{'} = z + c^{'} p

, where

i, j, z \in S^{'}

and

a^{'}, b^{'}, c^{'} \in {0, 1, . . ., q - 1}

. Since

G (B_{i}^{'})

is adjacent to

G (B_{j}^{'})

G_{p q}^{'}

and x is nonadjacent to y, then

(i + j, p) = 1

and

((i + j) + (a^{'} + b^{'}) p, p q) \neq 1

, this implies that

((i + j) + (a^{'} + b^{'}) p, q) \neq 1

. Similarly, since

G (B_{z}^{'})

is adjacent to

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

, then

((i + z^{'}) + (a^{'} + c^{'}) p, q) \neq 1

and

((j + z^{'}) + (b^{'} + c^{'}) p, q) \neq 1

. So,

\begin{matrix} x + y = (i + j) + (a^{'} + b^{'}) p = a_{1} q, \\ x + z^{'} = (i + z) + (a^{'} + c^{'}) p = a_{2} q, \\ y + z^{'} = (j + z) + (b^{'} + c^{'}) p = a_{3} q . \end{matrix}

So,

2 z^{'} = (a_{2} + a_{3} - a_{1}) q

; this implies that 2 divide

(a_{2} + a_{3} - a_{1})

, and hence

z^{'}

is multiple of q, in this case

z^{'} = z^{*}

. But

z^{*}

is adjacent to all vertices in

G (B_{i}^{'})

and

G (B_{j}^{'})

except

x = x^{*}

and

y = y^{*}

, respectively. This is a contradiction with

x \neq x^{*}

and

y \neq y^{*}

. So,

x = x^{*}

and

y = y^{*}

(\Leftarrow)

Assume that

x = x^{*}

and

y = y^{*}

. Then, x is adjacent to

(q - 1)

vertices of

G (B_{z}^{'})

by Part 2 of Lemma 6, that is x is adjacent to all vertices of

G (B_{z}^{'})

except

z^{*}

, where

z^{*} = c q

and

c \in {0, 1, . . ., q - 1}

. Similarly, y is adjacent to all vertices of

G (B_{z}^{'})

except

z^{*}

. Then, the result is obtained. □

5.2. Number of Internally Disjoint Paths between Nonadjacent Vertices in $G (Z_{p q})$

The following two lemmas calculate the number of internally disjoint paths between nonadjacent vertices x and y in

G (B_{i}^{'})

Lemma 11.

Let

x, y \in G (B_{i}^{'})

be nonadjacent. Then, there are

(p - 1) (q - 2)

internally disjoint paths of length 2 between x and y.

Proof.

i \neq 0

, then

G (B_{i}^{'})

is isomorphic to

K_{1} \lor C P (q - 1)

by Part 1 of Lemma 6. Since

x^{*}

represents

K_{1}

, then

x^{*}

adjacent to all vertices in

G (B_{i}^{'})

. Since x is nonadjacent to y, then both x and y are not

x^{*}

G (B_{i}^{'})

. So, there are

(q - 2)

common neighbors between x and y, and hence there are

(q - 2)

internally disjoint paths of length 2 between x and y in

G (B_{i}^{'})

. Let

G (B_{z}^{'})

be a neighbor of

G (B_{i}^{'})

G_{p q}^{'}

. By Lemma 9, there are

(q - 2)

common neighbors between x and y in

G (B_{z}^{'})

. Hence, there are

(q - 2)

internally disjoint paths of length 2 through

G (B_{z}^{'})

. By Lemma 7, there are

(p - 2)

neighbors of

G (B_{i}^{'})

G_{p q}^{'}

, and hence there are

(p - 2) (q - 2)

internally disjoint paths of length 2 between x and y through all neighbors of

G (B_{i}^{'})

G_{p q}^{'}

. Thus, the total number of internally disjoint paths of length 2 between x and y is

(q - 2) + (p - 2) (q - 2) = (p - 1) (q - 2) .

i = 0

, then

G (B_{0}^{'})

is isomorphic to

{\bar{K}}_{q}

, by Part 1 of Lemma 6. So, there is no path between x and y in

G (B_{0}^{'})

. By Lemma 9, there are

(q - 2)

common neighbors between x and y in

G (B_{z}^{'})

, where

G (B_{z}^{'})

is a neighbor of

G (B_{0}^{'})

G_{p q}^{'}

. Hence, there are

(q - 2)

internally disjoint paths of length 2 through

G (B_{z}^{'})

. By Lemma 7, there are

(p - 1)

neighbors of

G (B_{0}^{'})

G_{p q}^{'}

, and hence there are

(p - 1) (q - 2)

internally disjoint paths of length 2 between x and y. □

Lemma 12.

Let

x, y \in G (B_{i}^{'})

be nonadjacent. Then:

1.: If $i \neq 0$ , then there are $(p - 2)$ internally disjoint paths of length 4 between x and y.
2.: If $i = 0$ , then there are $(p - 1)$ internally disjoint paths of length 4 between x and y.

Proof.

Let

x, y \in G (B_{i}^{'})

be nonadjacent and

G (B_{z}^{'})

be a neighbor of

G (B_{i}^{'})

G_{p q}^{'}

. By proof of Lemma 9, x and y are adjacent to all vertices of

G (B_{z}^{'})

except

a_{z}

and

b_{z}

, respectively.

Let $i \neq 0$ . By Part 2 of Lemma 6, $a_{z}$ and $b_{z}$ in $G (B_{z}^{'})$ are adjacent to $(q - 1)$ vertices of $G (B_{p - i}^{'})$ . Since we investigate the internally disjoint paths between x and y through $a_{z}$ and $b_{z}$ , we can choose a vertex w from $G (B_{p - i}^{'})$ that is adjacent to both $a_{z}$ and $b_{z}$ . This path will be of length 4, as illustrated in Figure 4 (a). Similarly, for each neighbor $G (B_{z}^{'})$ of $G (B_{i}^{'})$ in $G_{p q}^{'}$ there is one internally disjoint path of length 4 between x and y. By Lemma 7, there are $(p - 2)$ neighbors of $G (B_{i}^{'})$ in $G_{p q}^{'}$ . Therefore, there are $(p - 2)$ internally disjoint paths of length 4 between x and y through all neighbors of $G (B_{i}^{'})$ .
Let $i = 0$ . By Part 2 of Lemma 6, $a_{z}$ and $b_{z}$ in $G (B_{z}^{'})$ are adjacent to $(q - 1)$ vertices of $G (B_{0}^{'})$ . Approaching the proof in a similar manner as with Part 1, there is a path of length 4, as shown in Figure 4 (b). By Lemma 7, the number of neighbors of $G (B_{0}^{'})$ in $G_{p q}^{'}$ is $(p - 1)$ . So, there are $(p - 1)$ internally disjoint paths of length 4 between x and y through all neighbors of $G (B_{0}^{'})$ . □

The following two lemmas determine the number of internally disjoint paths between nonadjacent vertices

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

, where

G (B_{i}^{'})

is nonadjacent to

G (B_{j}^{'})

G_{p q}^{'}

Lemma 13.

Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

be nonadjacent,

G (B_{i}^{'})

be nonadjacent to

G (B_{j}^{'})

G_{p q}^{'}

, and

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. Then:

1.: If x and y have the same neighbors in $G (B_{z}^{'})$ , then there are $(p - 2) (q - 1)$ internally disjoint paths of length 2 between x and y.
2.: If x and y do not have the same neighbors in $G (B_{z}^{'})$ , then there are $(p - 2) (q - 2)$ internally disjoint paths of length 2 between x and y.

Proof.

Let

G (B_{i}^{'})

be nonadjacent to

G (B_{j}^{'})

G_{p q}^{'}

. By Part 1 of Lemma 8, there are

(p - 2)

common neighbors between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. Then:

If x and y have the same neighbors in $G (B_{z}^{'})$ , then there are $(q - 1)$ common neighbors between x and y in $G (B_{z}^{'})$ by Lemma 10. Thus, there are $(q - 1)$ internally disjoint paths of length 2 between x and y through $G (B_{z}^{'})$ . Hence, there are $(p - 2) (q - 1)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ .
If x and y do not have the same neighbors in $G (B_{z}^{'})$ , then there are $(q - 2)$ common neighbors between x and y in $G (B_{z}^{'})$ by Lemma 10. So, there are $(q - 2)$ internally disjoint paths of length 2 between x and y through $G (B_{z}^{'})$ . Therefore, there are $(p - 2) (q - 2)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ . □

Lemma 14.

Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

be nonadjacent,

G (B_{i}^{'})

be nonadjacent to

G (B_{j}^{'})

G_{p q}^{'}

, and

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. Then:

1.: If x and y have the same neighbors in $G (B_{z}^{'})$ , then there are $(p - 2)$ internally disjoint paths of length 4 between x and y.
2.: If x and y do not have the same neighbors in $G (B_{z}^{'})$ , then there are $2 (p - 2)$ internally disjoint paths of length 3 between x and y.

Proof.

Let

G (B_{i}^{'})

be nonadjacent to

G (B_{j}^{'})

G_{p q}^{'}

. By Part 1 of Lemma 8, there are

(p - 2)

common neighbors between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

Let x and y have the same neighbors in $G (B_{z}^{'})$ . By Lemma 6, $x \in G (B_{i}^{'})$ has at least $(q - 2)$ neighbors in $G (B_{i}^{'})$ and each of them is adjacent to $(q - 1)$ vertices of $G (B_{z}^{'})$ . Then, we can choose a neighbor, say $x^{'}$ , of x in $G (B_{i}^{'})$ such that $x^{'} \sim z^{'}$ , where $z^{'} \in G (B_{z}^{'})$ is nonadjacent to both x and y. Similarly, we can choose a neighbor $y^{'}$ of y in $G (B_{j}^{'})$ such that $y^{'} \sim z^{'}$ . Therefore, there exists a path of length 4 of the form $x - x^{'} - z^{'} - y^{'} - y$ , see Figure 5 (a). So, there are $(p - 2)$ internally disjoint paths of length 4 between x and y through all common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ .
Let x and y do not have the same neighbors in $G (B_{z}^{'})$ . By proof of Lemma 10, x and y are adjacent to all vertices in $G (B_{z}^{'})$ except $a_{z}$ and $b_{z}$ , respectively. Approaching the proof similarly as Part 1, we can choose a neighbor $x^{'}$ of x in $G (B_{i}^{'})$ and a neighbor $y^{'}$ of y in $G (B_{j}^{'})$ , where $x^{'} \sim a_{z}$ and $y^{'} \sim b_{z}$ . Since $a_{z} \sim y$ and $b_{z} \sim x$ , two internally disjoint paths of length 3 exist. These paths are described in Figure 5 (b). Hence, there are $2 (p - 2)$ internally disjoint paths of length 3 between x and y through all common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ . □

The following results find the number of internally disjoint paths between nonadjacent vertices

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

, where

G (B_{i}^{'})

is adjacent to

G (B_{j}^{'})

G_{p q}^{'}

Lemma 15.

Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

be nonadjacent and

G (B_{i}^{'})

be adjacent to

G (B_{j}^{'})

G_{p q}^{'}

. One of the following cases holds:

1.: There are $(p - 2) (q - 1)$ internally disjoint paths of length 2 between x and y.
2.: There are $(p - 2) (q - 2)$ internally disjoint paths of length 2 between x and y.

Proof.

By Lemma 8, there are

(p - 4)

common neighbors between the adjacent vertices

G (B_{i}^{'})

and

G (B_{j}^{'})

, where

i, j \neq 0

, in

G_{p q}^{'}

and there are

(p - 3)

common neighbors between

G (B_{0}^{'})

and

G (B_{i}^{'})

G_{p q}^{'}

. Let

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. Since x and y are nonadjacent, then x and y have the following probabilities:

Assume that x and y have the same neighbors in $G (B_{z}^{'})$ . By Proposition 2 and Lemma 10, $x = x^{*}$ and $y = y^{*}$ and there are $(q - 1)$ common neighbors between x and y through $G (B_{z}^{'})$ . So, there are $(q - 1)$ internally disjoint paths of length 2 between x and y through $G (B_{z}^{'})$ . If $i, j \neq 0$ , then there are $(p - 4) (q - 1)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ . Further, x (resp. y) is adjacent to all vertices in $G (B_{i}^{'})$ (resp. $G (B_{j}^{'})$ ). Also, x (resp. y) is adjacent to all vertices in $G (B_{j}^{'})$ (resp. $G (B_{i}^{'})$ ) except y (resp. x). So, the set of common neighbors between x and y in $G (B_{i}^{'})$ and $G (B_{j}^{'})$ is $G (B_{i}^{'}) - {x}$ union $G (B_{j}^{'}) - {y}$ . Thus, there are $2 (q - 1)$ common neighbors between x and y in $G (B_{i}^{'})$ and $G (B_{j}^{'})$ . Consequently, there are $2 (q - 1)$ internally disjoint paths of length 2 between x and y in $G (B_{i}^{'})$ and $G (B_{j}^{'})$ . Therefore, the total number of internally disjoint paths of length 2 between x and y is

$(p - 4) (q - 1) + 2 (q - 1) = (p - 2) (q - 1) .$

If $j = 0$ , then there are $(p - 3) (q - 1)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G (B_{0}^{'})$ and $G (B_{i}^{'})$ in $G_{p q}^{'}$ . Further, x is adjacent to all vertices in $G (B_{i}^{'})$ and y is nonadjacent to any vertex in $G (B_{0}^{'})$ . Also, x (resp. y) is adjacent to all vertices in $G (B_{0}^{'})$ (resp. $G (B_{i}^{'})$ ) except y (resp. x). So, the set of common neighbors between x and y in $G (B_{0}^{'})$ and $G (B_{i}^{'})$ are $G (B_{i}^{'}) - {x}$ . Thus, there are $(q - 1)$ common neighbors between x and y in $G (B_{0}^{'})$ and $G (B_{i}^{'})$ . Then, there are $(q - 1)$ internally disjoint paths of length 2 between x and y in $G (B_{0}^{'})$ and $G (B_{i}^{'})$ . Thus, the total number of internally disjoint paths of length 2 between x and y is

$(p - 3) (q - 1) + (q - 1) = (p - 2) (q - 1) .$
Assume that x and y do not have the same neighbors in $G (B_{z}^{'})$ . By Proposition 2 and Lemma 10, $x \neq x^{*}$ and $y \neq y^{*}$ and there are $(q - 2)$ common neighbors between x and y in $G (B_{z}^{'})$ . Thus, there are $(q - 2)$ internally disjoint paths of length 2 between x and y through $G (B_{z}^{'})$ . If $i, j \neq 0$ , then there are $(p - 4) (q - 2)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ . Furthermore, x (resp. y) is adjacent to all vertices except only one vertex $x^{'}$ (resp. $y^{'}$ ) in $G (B_{i}^{'})$ (resp. $G (B_{j}^{'})$ ). Also, x (resp. y) is adjacent to all vertices in $G (B_{j}^{'})$ (resp. $G (B_{i}^{'})$ ) except y (resp. x). Thus, the set of common neighbors between x and y in $G (B_{i}^{'})$ and $G (B_{j}^{'})$ is $G (B_{i}^{'}) - {x, x^{'}}$ union $G (B_{j}^{'}) - {y, y^{'}}$ . So, there are $2 (q - 2)$ common neighbors between x and y in $G (B_{i}^{'})$ and $G (B_{j}^{'})$ . As a result, there are $2 (q - 2)$ internally disjoint paths of length 2 between x and y in $G (B_{i}^{'})$ and $G (B_{j}^{'})$ . Therefore, the total number of internally disjoint paths of length 2 between x and y is

$(p - 4) (q - 2) + 2 (q - 2) = (p - 2) (q - 2) .$

If $j = 0$ , then there are $(p - 3) (q - 2)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G (B_{0}^{'})$ and $G (B_{i}^{'})$ in $G_{p q}^{'}$ . Further, x is adjacent to all vertices except only one vertex $x^{'}$ in $G (B_{i}^{'})$ , and y is nonadjacent to any vertex in $G (B_{0}^{'})$ . Also, x (resp. y) is adjacent to all vertices in $G (B_{0}^{'})$ (resp. $G (B_{i}^{'})$ ) except y (resp. x). Thus, the set of common neighbors between x and y in $G (B_{0}^{'})$ and $G (B_{i}^{'})$ is $G (B_{i}^{'}) - {x, x^{'}}$ . So, there are $(q - 2)$ common neighbors between x and y in $G (B_{0}^{'})$ and $G (B_{i}^{'})$ . Then, there are $(q - 2)$ internally disjoint paths of length 2 between x and y in $G (B_{0}^{'})$ and $G (B_{i}^{'})$ . Thus, the total number of internally disjoint paths of length 2 between x and y is

$(p - 3) (q - 2) + (q - 2) = (p - 2) (q - 2) .$

□

Lemma 16.

Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

be nonadjacent, where

G (B_{i}^{'})

is adjacent to

G (B_{j}^{'})

G_{p q}^{'}

. One of the following cases holds:

1.: There are $(q - 1)$ internally disjoint paths of length 3 between x and y.
2.: There are $(p + q - 5)$ internally disjoint paths of length 3 between x and y.
3.: There are $(p + q - 4)$ internally disjoint paths of length 3 between x and y.

Proof.

We need to examine whether any

x y

-path passes through

G (B_{p - j}^{'})

and

G (B_{p - i}^{'})

because we are sure that

G (B_{p - j}^{'})

(resp.

G (B_{p - i}^{'})

) is adjacent to

G (B_{i}^{'})

(resp.

G (B_{j}^{'})

) and nonadjacent to

G (B_{j}^{'})

(resp.

G (B_{i}^{'})

) in

G_{p q}^{'}

. Let

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. Since x and y are nonadjacent, then x and y have the following cases:

Assume that x and y have the same neighbors in $G (B_{z}^{'})$ . Indeed, $x = x^{*}$ and $y = y^{*}$ by Proposition 2. If $i, j \neq 0$ , there are $(q - 1)$ neighbors of x in $G (B_{p - j}^{'})$ , denote these neighbors by $x_{k}^{'}$ such that $k = 1, 2, . . ., q - 1$ , and each of them is adjacent to $(q - 1)$ vertices of $G (B_{p - i}^{'})$ by Part 2 of Lemma 6. Similarly, there are $(q - 1)$ neighbors of y in $G (B_{p - i}^{'})$ , and each of them is adjacent to $(q - 1)$ vertices of $G (B_{p - j}^{'})$ . To get the internally disjoint paths of length 3 between x and y, we choose one of the neighbors of $x_{k}^{'}$ , say $y_{k}^{'}$ , in $G (B_{p - i}^{'})$ such that $y_{k}^{'}$ is a neighbor of y. Indeed, for each $x_{k}^{'}$ in $G (B_{p - j}^{'})$ there is one internally disjoint path between x and y through $G (B_{p - j}^{'})$ and $G (B_{p - i}^{'})$ . Therefore, the total number of internally disjoint paths of length 3 between x and y through $G (B_{p - j}^{'})$ and $G (B_{p - i}^{'})$ together is equal to the number of neighbors of x in $G (B_{p - j}^{'})$ , which is $(q - 1)$ . Now let $j = 0$ . There are $(q - 1)$ neighbors of x in $G (B_{0}^{'})$ and each of them is adjacent to $(q - 1)$ vertices of $G (B_{p - i}^{'})$ . Since there are $(q - 1)$ neighbors of y in $G (B_{p - i}^{'})$ , so there are more than $(q - 1)$ paths of length 3 between x and y through $G (B_{0}^{'})$ and $G (B_{p - i}^{'})$ together. By applying the same method in the case where $i, j \neq 0$ , there are $(q - 1)$ internally disjoint paths of length 3 between x and y.
Assume that x and y do not have the same neighbors in $G (B_{z}^{'})$ . So, $x \neq x^{*}$ and $y \neq y^{*}$ by Proposition 2. Suppose that $G (B_{z_{k}}^{'})$ is a common neighbor between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ . By proof of Lemma 10, x and y are adjacent to all vertices in $G (B_{z_{k}}^{'})$ except $a_{z_{k}}$ and $b_{z_{k}}$ , respectively. Let $i, j \neq 0$ . Since x has $(q - 1)$ neighbors in $G (B_{p - j}^{'})$ and each of these neighbors is adjacent to $(q - 1)$ vertices of $G (B_{z_{k}}^{'})$ , then we can choose a neighbor $x_{k}^{'}$ of x in $G (B_{p - j}^{'})$ such that $x_{k}^{'} \sim a_{z_{K}}$ . Similarly, we can choose a neighbor $y_{k}^{'}$ of y in $G (B_{p - i}^{'})$ such that $y_{k}^{'} \sim b_{z_{K}}$ . Since $a_{z_{k}} \sim y$ and $b_{z_{k}} \sim x$ , there exist two internally disjoint paths of length 3 between x and y, as illustrated in Figure 6 (a). By Part 2 of Lemma 8, there are $(p - 4)$ common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ . Then, there are $2 (p - 4)$ internally disjoint paths of length 3 between x and y through all common neighbors between $G (B_{i}^{'})$ and $G (B_{j}^{'})$ in $G_{p q}^{'}$ . After removing all $x_{k}^{'}$ and $y_{k}^{'}$ from $G (B_{p - j}^{'})$ and $G (B_{p - i}^{'})$ , respectively, then the number of remaining neighbors of x and y in $G (B_{p - j}^{'})$ and $G (B_{p - i}^{'})$ , respectively, is $(q - 1) - (p - 4) = q - p + 3$ . So, there are $(q - p + 3)$ internally disjoint paths length 3 between x and y that pass through the remaining of neighbors of x and y in $G (B_{p - j}^{'})$ and $G (B_{p - i}^{'})$ , respectively, together. So, the total number of internally disjoint paths of length 3 between x and y is

$2 (p - 4) + (q - p + 3) = p + q - 5 .$

Let $j = 0$ . Since x has $(q - 1)$ neighbors in $G (B_{0}^{'})$ and each of these neighbors is adjacent to $(q - 1)$ vertices of $G (B_{z_{k}}^{'})$ , then we can choose a neighbor $x_{k}^{'}$ of x in $G (B_{0}^{'})$ such that $x_{k}^{'} \sim a_{z_{k}}$ . Similarly, we can choose a neighbor $y_{k}^{'}$ of y in $G (B_{p - i}^{'})$ such that $y_{k}^{'} \sim b_{z_{k}}$ . Since $a_{z_{k}} \sim y$ and $b_{z_{k}} \sim x$ , there exist two internally disjoint path of length 3 between x and y, as illustrated in Figure 6 (b). By Part 3 of Lemma 8, there are $(p - 3)$ common neighbors between $G (B_{0}^{'})$ and $G (B_{i}^{'})$ in $G_{p q}^{'}$ . Consequently, there are $2 (p - 3)$ internally disjoint paths of length 3 between x and y through all common neighbors between $G (B_{0}^{'})$ and $G (B_{i}^{'})$ in $G_{p q}^{'}$ . After removing all $x_{k}^{'}$ and $y_{k}^{'}$ from $G (B_{0}^{'})$ and $G (B_{p - i}^{'})$ , respectively, then the number of remaining neighbors of x and y in $G (B_{0}^{'})$ and $G (B_{p - i}^{'})$ , respectively, is $(q - 1) - (p - 3) = q - p + 2$ . So, there are $(q - p + 2)$ internally disjoint paths length 3 between x and y that pass through the rest of neighbors of x and y in $G (B_{0}^{'})$ and $G (B_{p - i}^{'})$ , respectively, together. So, the total number of internally disjoint paths of length 3 between x and y is

$2 (p - 3) + (q - p + 2) = p + q - 4 .$

□

5.3. Vertex Connectivity of $G (Z_{p^{r} q^{s}})$

The following result is of crucial importance to our study in this section.

Theorem 6.

Let

2 \neq p < q

be distinct primes. The vertex connectivity of

G (Z_{p q})

κ (G (Z_{p q})) = (p - 2) q .

Proof.

Let x and y be nonadjacent in

G (Z_{p q})

. In this proof, we will calculate the maximum number of internally disjoint paths between any two nonadjacent vertices. There are several cases for x and y, as follows:

Case 1: Let

x, y \in G (B_{i}^{'})

. By Lemma 11, then there are

(p - 1) (q - 2)

internally disjoint paths of length 2 between x and y. In addition, there are other internally disjoint paths depending on the following cases for i:

(a): Let $i \neq 0$ . By lemma 12, there are $(p - 2)$ internally disjoint paths of length 4 between x and y. So, the maximum number of internally disjoint paths between x and y is

$(p - 1) (q - 2) + (p - 2) = p q - p - q .$
(b): Let $i = 0$ . By Lemma 12, there are $(p - 1)$ internally disjoint paths of length 4 between x and y. So, the maximum number of internally disjoint paths between x and y is

$(p - 1) (q - 2) + (p - 1) = (p - 1) (q - 1) .$

Case 2: Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

, where

G (B_{i}^{'})

is nonadjacent to

G (B_{j}^{'})

G_{p q}^{'}

. Let

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. The following cases arise for x and y:

(a): If y has the same neighbors as x in $G (B_{z}^{'})$ , then there are $(p - 2) (q - 1)$ internally disjoint paths of length 2 between x and y by Lemma 13. According to Lemma 14, there are $(p - 2)$ internally disjoint paths of length 4 between x and y. Hence, the maximum number of internally disjoint paths between x and y is

$(p - 2) (q - 1) + (p - 2) = (p - 2) q .$
(b): If x and y do not have the same neighbors in $G (B_{z}^{'})$ , there are $(p - 2) (q - 2)$ internally disjoint paths of length 2 between x and y by Lemma 13. According to Lemma 14, there are $2 (p - 2)$ internally disjoint paths of length 3 between x and y. Hence, the maximum number of internally disjoint paths between x and y is

$(p - 2) (q - 2) + 2 (p - 2) = (p - 2) q .$

Case 3: Let

x \in G (B_{i}^{'})

and

y \in G (B_{j}^{'})

, where

G (B_{i}^{'})

is adjacent to

G (B_{j}^{'})

G_{p q}^{'}

. Let

G (B_{z}^{'})

be a common neighbor between

G (B_{i}^{'})

and

G (B_{j}^{'})

G_{p q}^{'}

. There are the following cases for x and y:

(a)

If y has the same neighbors as x in

G (B_{z}^{'})

, then there are

(p - 2) (q - 1)

internally disjoint paths of length 2 between x and y by proof of Lemma 15. According to proof of Lemma 16, there are

(q - 1)

internally disjoint paths of length 3 between x and y. Hence, the maximum number of internally disjoint paths between x and y is

(p - 2) (q - 1) + (q - 1) = (p - 1) (q - 1) .

(b)

If x and y do not have the same neighbors in

G (B_{z}^{'})

, then there are

(p - 2) (q - 2)

internally disjoint paths of length 2 between x and y by proof of Lemma 15. In addition, there are other internally disjoint paths depending on the following cases for i and j:

(1): Let $i, j \neq 0$ . According to proof of Lemma 16, there are $(p + q - 5)$ internally disjoint paths of length 3 between x and y. Therefore, the maximum number of internally disjoint paths between x and y is

$(p - 2) (q - 2) + p + q - 5 = p q - p - q - 1 .$
(2): Let $j = 0$ . According to proof of Lemma 16, there are $(p + q - 4)$ internally disjoint paths of length 3 between x and y. So, the maximum number of internally disjoint paths between x and y is

$(p - 2) (q - 2) + (p + q - 4) = p q - p - q .$

From the above cases and by Menger’s theorem, we have

\begin{matrix} κ (G (Z_{p q})) & = min {p q - p - q, (p - 1) (q - 1), (p - 2) q, p q - p - q - 1} \\ = (p - 2) q . \end{matrix}

□

Now, let us explore the vertex connectivity of

G (Z_{n})

n = p^{r} q^{s}

, where

2 \neq p < q

are primes and

r, s

are positive integers such that at least one of

r, s

must be greater than 1.

Theorem 7.

Let

n = p^{r} q^{s}

, where

2 \neq p < q

are distinct primes, r and s are positive integers. Then, the vertex connectivity of

G (Z_{n})

is given by

κ (G (Z_{n})) = (p - 2) p^{(r - 1)} q^{s} .

Proof.

By Lemma 4, the unit graph

G (Z_{n})

G (Z_{n}) = G_{p^{r} q^{s}} [G (B_{0}), G (B_{1}), \dots, G (B_{p q - 1})] .

According to Lemma 3,

G_{p^{r} q^{s}}

is isomorphic to

G (Z_{p q})

. Hence, by Theorem 6, we get

κ (G_{p^{r} q^{s}}) = (p - 2) q .

Note that, for every vertex i of

G_{p^{r} q^{s}}

, we have

| G (B_{i}) |

vertices in

G (Z_{n})

. Since

| G (B_{i}) | = p^{r - 1} q^{s - 1}

for

0 \leq i \leq p q - 1

, then the vertex connectivity of

G (Z_{n})

\begin{matrix} κ (G (Z_{n})) & = p^{r - 1} q^{s - 1} κ (G_{p^{r} q^{s}}) \\ = (p - 2) p^{(r - 1)} q^{s} . \end{matrix}

□

Based on our results for the vertex connectivity of the unit graph, in Theorems 6 and 7, we state the following conjectures:

Conjecture I: Let

n = p_{1} p_{2} . . . p_{k}

, where

2 \neq p_{1} < p_{2} < . . . < p_{k}

are distinct primes. The vertex connectivity of

G (Z_{n})

κ (G (Z_{n})) = (p_{1} - 2) \prod_{2 \leq i \leq k} p_{i} .

Conjecture II: Let

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

2 \neq p_{1} < p_{2} < . . . < p_{k}

are distinct primes,

r_{i}

and k are positive integers, and

1 \leq i \leq k

. Then, the vertex connectivity of

G (Z_{n})

is given by

κ (G (Z_{n})) = (p_{1} - 2) p_{1}^{(r_{1} - 1)} \prod_{2 \leq i \leq k} p_{i}^{r_{i}} .

6. Conclusions

In this paper, we have investigated the structure of

G (Z_{n})

. Based on this structure, the Laplacian spectrum and vertex connectivity of

G (Z_{n})

have been determined for various n. First, we study the structure of

G (Z_{n})

for

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

p_{1}, p_{2}, . . ., p_{k}

are distinct primes and

k, r_{1}, r_{2}, . . ., r_{k}

are positive integers such that at least one of the

r_{i}

must be greater than 1, and we prove that the graph

G (Z_{n})

is a generalized join of certain complete graphs and null graphs. Then, we determine the Laplacian spectrum of

G (Z_{n})

, we prove that

G (Z_{n})

is Laplacian integral, and we deduce the algebraic connectivity and Laplacian spectral radius of

G (Z_{n})

. Furthermore, we examine the vertex connectivity of

G (Z_{p q})

and

G (Z_{p^{r} q^{s}})

, where

2 \neq p < q

are primes and r and s are positive integers by using their structure and Menger’s theorem. Finally, we present conjectures about the vertex connectivity of

G (Z_{n})

when

n = p_{1} p_{2} . . . p_{k}

and

n = p_{1}^{r_{1}} p_{2}^{r_{2}} . . . p_{k}^{r_{k}}

, where

p_{i}

are distinct primes,

r_{i}

are positive integers, and

1 \leq i \leq k

. Our results are precise and dependable, as verified by Python programming (see Appendix A).

Author Contributions

Investigation, A. A. and W. F.; Methodology, A. A. and W. F.; Writing—–original draft preparation, A. A., W. F. and H. A.; Writing–—review and editing, A. A., W. F. and H. A; ; supervision, W. F. and H. A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

Pranjali et al. [19] provided the generation code of the unit graph of

Z_{n}

. We utilize this code in Python programming to create the following algorithm that verifies the validity of our results.

References

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1	$C P (q - 1)$ is the cocktail party graph, which is obtained from the complete graph $K_{2 s}$ , $2 s = q - 1$ , by deleting a perfect matching, where a perfect matching of graph G is a 1-regular spanning subgraph H of G.

Figure 1. The graph

G (Z_{25}

Figure 1. The graph

G (Z_{25}

Figure 2. The graph

G_{25}

Figure 2. The graph

G_{25}

Figure 3. The graph

G (Z_{15})

Figure 3. The graph

G (Z_{15})

Figure 4. (a)

x - b_{z} - w - a_{z} - y

when

i \neq 0

. (b)

x - b_{z} - w - a_{z} - y

when

i = 0

Figure 4. (a)

x - b_{z} - w - a_{z} - y

when

i \neq 0

. (b)

x - b_{z} - w - a_{z} - y

when

i = 0

Figure 5. (a)

x - x^{'} - z^{'} - y^{'} - y

. (b)

x - x^{'} - a_{z} - y and x - b_{z} - y^{'} - y .

Figure 5. (a)

x - x^{'} - z^{'} - y^{'} - y

. (b)

x - x^{'} - a_{z} - y and x - b_{z} - y^{'} - y .

Figure 6. (a)

x - x_{k}^{'} - a_{z_{k}} - y

and

x - b_{z_{k}} - y_{k}^{'} - y

when

i, j \neq 0

. (b)

x - x_{k}^{'} - a_{z_{k}} - y

and

x - b_{z_{k}} - y_{k}^{'} - y

when

j = 0

Figure 6. (a)

x - x_{k}^{'} - a_{z_{k}} - y

and

x - b_{z_{k}} - y_{k}^{'} - y

when

i, j \neq 0

. (b)

x - x_{k}^{'} - a_{z_{k}} - y

and

x - b_{z_{k}} - y_{k}^{'} - y

when

j = 0

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring Zprqs

Abstract

Keywords:

Subject:

1. Introduction

2. Preliminaries

3. G ( Z n ) as a Generalized Join Graph

4. Laplacian Spectrum of G ( Z n )

5. Vertex Connectivity of G ( Z n )

5.1. Structure of G ( Z p q )

5.2. Number of Internally Disjoint Paths between Nonadjacent Vertices in G ( Z p q )

5.3. Vertex Connectivity of G ( Z p r q s )

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A

References

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3. $G (Z_{n})$ as a Generalized Join Graph

4. Laplacian Spectrum of $G (Z_{n})$

5. Vertex Connectivity of $G (Z_{n})$

5.1. Structure of $G (Z_{p q})$

5.2. Number of Internally Disjoint Paths between Nonadjacent Vertices in $G (Z_{p q})$

5.3. Vertex Connectivity of $G (Z_{p^{r} q^{s}})$