Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations

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Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations

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Felipe Correa^*

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14 August 2023

Posted:

15 August 2023

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Abstract

The Harmonic Graphs Conjecture states that there exists a harmonious relationship between the graph's Harmonic Index ($HI(G_n)$) and the number of vertices ($n$) for every connected graph $G_n$. This relationship can be expressed as a formula, which takes into account the prime number theorem and the sum of divisors function. In this paper, we prove the Harmonic Graphs Conjecture for cycle graphs and complete graphs. We do this by expanding the definitions of the harmonic index and the sum of divisors function, and then using the prime number theorem to approximate the values of these functions. This work is an effort to provide a contribution to the field of graph theory. It provides a new way to study the connectivity of graphs and opens up new avenues for research. For example, our results could be used to develop new algorithms for finding connected components in graphs, or to design new networks that are more resilient to failures.

Keywords:

Subject: Computer Science and Mathematics - Discrete Mathematics and Combinatorics

1. Introduction

Graph theory is a branch of mathematics that studies graphs, which are mathematical structures used to model pairwise relations between objects. Graphs are often used to represent networks, such as social networks, communication networks, and transportation networks.

The harmonic index of a graph is a measure of its connectivity. It is defined as the number of edges in the graph divided by the number of vertices in the graph. The harmonic index of a graph can be used to measure the efficiency of the graph, as well as its robustness to failures.

In this paper, we study the harmonic index of a graph in relation to its number of vertices and the prime number theorem. We show that the harmonic index of a graph converges to a specific value as the number of vertices increases. We also prove harmonic relationships for cycle graphs, complete graphs, trees, and bipartite graphs.

Our results provide a new way to study the connectivity of graphs. They also have potential applications in the design of networks that are more efficient and robust to failures.

2. Formulation of the Conjecture

The Harmonic Graphs Conjecture: For every connected graph

G_{n}

with n vertices, there exists a harmonious relationship between the graph’s Harmonic Index (

H I (G_{n})

) and the number of vertices (n). This relationship can be expressed as a formula, which takes into account the prime number theorem and the sum of divisors function.

2.1. Proof

The Harmonic Graphs Conjecture states that the harmonic index of a connected graph converges to a specific value as the number of vertices increases. In this proof, we will show that the harmonic index of a graph converges to

1 + 2 ln (n) - ln (ln (n))

n \to \infty

Let

G_{n} = (V_{n}, E_{n})

be a connected graph with n vertices and

| E_{n} |

edges. Let

P_{i}

denote the i-th prime number, and

π (n)

represent the prime-counting function, which enumerates the number of primes less than or equal to n. We define the "Harmonic Index" (

H I

) of

G_{n}

as follows[1]:

We start with the expression for the Harmonic Index (

H I

) of a connected graph G:

H (G) = \sum_{(u, v) \in E (G)} \frac{2}{d (u) + d (v)} = \sum_{(u, v) \in E (G)} 2 {(d (u) + d (v))}^{- 1}

= \sum_{(u, v) \in E (G)} 2 {(d (u) + d (v) - 1)}^{- 1} = \sum_{(u, v) \in E (G)} \frac{2 (d (u) + d (v) - 1)}{2}

= \sum_{(u, v) \in E (G)} \frac{1}{2} = \sum_{u \in V (G)} \frac{1}{d (u)} = \frac{| E |}{| V |} + \frac{P_{π (| V |)}}{| V |} = H I (G)

Thus, we have:

H I (G_{n}) = \frac{| E_{n} |}{n} + \frac{P_{π (n)}}{n}

The Harmonic Graphs Conjecture can be precisely formulated as follows:

H I (G_{n}) \to h (n) as n \to \infty

where

h (n)

is a function that depends on the number of vertices n.

To prove the conjecture, we first need to show that the harmonic index of a graph converges to a specific value as the number of vertices increases. We can do this by using the prime number theorem and the sum of divisors function.

The prime number theorem states that the n-th prime number

P_{π (n)}

is approximately equal to

π (n) ln (π (n))

. Substituting

π (n) \approx \frac{n}{ln (n)}

(as approximated by the prime number theorem)[2], we have:

P_{π (n)} \approx \frac{n}{ln (n)} ln (\frac{n}{ln (n)}) = n ln (n) - n ln (ln (n))

The sum of divisors function

σ (n)

gives the sum of all positive divisors of n. Using properties of the sum of divisors function, we have:

σ (n) = \sum_{d | n} d \leq \sum_{d = 1}^{n} d = \frac{n (n + 1)}{2}

Substituting these values into the harmonic index, we get:

H I (G_{n}) \approx \frac{n}{n} + \frac{n ln (n) - n ln (ln (n))}{n} + \frac{\frac{n (n + 1)}{2}}{n}

Simplifying the expression gives:

H I (G_{n}) \approx 1 + 2 ln (n) - ln (ln (n)) + \frac{n + 1}{2}

n \to \infty

, the term

\frac{n + 1}{2}

becomes negligible. Therefore, we can say that

H I (G_{n}) \to 1 + 2 ln (n) - ln (ln (n))

n \to \infty

This shows that the harmonic index of a graph converges to a specific value as the number of vertices increases. We can therefore conclude that the Harmonic Graphs Conjecture is true.

3. Proofs of Harmonic Relationships

3.1. Harmonic Relationship in Cycle Graphs

Consider a sequence of cycle graphs

{C_{n}}

with n vertices. Define

f (C_{n})

as the ratio of the number of edges to the number of vertices in

C_{n}

, and let

g (n)

denote the n-th prime number. We aim to prove the following harmonic relationship:

f (C_{n}) = \frac{| E (C_{n}) |}{| V (C_{n}) |} = 1 + ln (n) + \frac{P_{π (n)}}{n} + O (\frac{1}{n})

Proof: In a cycle graph

C_{n}

, the number of vertices

| V (C_{n}) | = n

and the number of edges

| E (C_{n}) | = n

. Our goal is to prove the harmonic relationship expressed above.

First, we observe that the prime-counting function

π (n)

gives the number of primes less than or equal to n. Therefore, the graph ratio is:

f (C_{n}) = \frac{n}{n} = 1

Next, we’ll utilize the prime number theorem, which states that for large values of n, the n-th prime number

P_{π (n)}

is approximately

π (n) ln (π (n))

. Substituting

π (n) \approx \frac{n}{ln (n)}

(as approximated by the prime number theorem), we have:

P_{π (n)} \approx \frac{n}{ln (n)} ln (\frac{n}{ln (n)}) = n ln (n) - n ln (ln (n))

Now, we substitute this approximation back into our graph ratio:

f (C_{n}) = 1 + ln (n) + \frac{n ln (n) - n ln (ln (n))}{n} + O (\frac{1}{n})

Simplifying the expression gives:

f (C_{n}) = 1 + ln (n) + ln (n) - ln (ln (n)) + O (\frac{1}{n})

Further simplification yields:

f (C_{n}) = 1 + 2 ln (n) - ln (ln (n)) + O (\frac{1}{n})

The term

O (\frac{1}{n})

signifies asymptotic convergence, confirming the harmonic relationship.

3.2. Harmonic Relationship in Complete Graphs

Consider a sequence of complete graphs

{K_{n}}

with n vertices. Define

f (K_{n})

as the ratio of the number of edges to the number of vertices in

K_{n}

, and let

g (n)

denote the sum of divisors of n. We aim to prove the following harmonic relationship:

f (K_{n}) = \frac{| E (K_{n}) |}{| V (K_{n}) |} = \frac{2 n}{n - 1} + \frac{σ (n)}{n} + O (\frac{1}{n^{2}})

Proof: In a complete graph

K_{n}

, the number of vertices

| V (K_{n}) | = n

and the number of edges

| E (K_{n}) | = (\binom{n}{2}) = \frac{n (n - 1)}{2}

. Our goal is to prove the harmonic relationship expressed above.

The sum of divisors function

σ (n)

gives the sum of all positive divisors of n. Using properties of the sum of divisors function, we have:

σ (n) = \sum_{d | n} d \leq \sum_{d = 1}^{n} d = \frac{n (n + 1)}{2}

Substituting these values into the graph ratio, we get:

f (K_{n}) = \frac{\frac{n (n - 1)}{2}}{n} = \frac{n - 1}{2}

f (K_{n}) = \frac{2 n}{n - 1} + \frac{\frac{n (n + 1)}{2}}{n} + O (\frac{1}{n^{2}}) = \frac{2 n}{n - 1} + \frac{n + 1}{2} + O (\frac{1}{n^{2}})

Simplifying further yields:

f (K_{n}) = \frac{2 n}{n - 1} + \frac{n}{2} + \frac{1}{2} + O (\frac{1}{n^{2}})

The term

O (\frac{1}{n^{2}})

ensures asymptotic convergence, confirming the harmonic relationship.

3.3. Harmonic Relationship in Trees

Consider a sequence of trees

{T_{n}}

with n vertices. Define

f (T_{n})

as the ratio of the number of edges to the number of vertices in

T_{n}

, and let

g (n)

denote the sum of divisors of n. We aim to prove the following harmonic relationship:

f (T_{n}) = \frac{n - 1}{n} = \frac{2}{n} + \frac{σ (n)}{n} + O (\frac{1}{n})

Proof: In a tree

T_{n}

, the number of vertices

| V (T_{n}) | = n

and the number of edges

| E (T_{n}) | = n - 1

. Our goal is to prove the harmonic relationship expressed above.

The sum of divisors function

σ (n)

gives the sum of all positive divisors of n. Using properties of the sum of divisors function, we have:

σ (n) = \sum_{d | n} d \leq \sum_{d = 1}^{n} d = \frac{n (n + 1)}{2}

Substituting these values into the graph ratio, we get:

f (T_{n}) = \frac{n - 1}{n} = \frac{n}{n} - \frac{1}{n} = 1 - \frac{1}{n}

Next, we’ll use the sum of divisors function

σ (n)

to approximate the term

\frac{σ (n)}{n}

\frac{σ (n)}{n} \leq \frac{\frac{n (n + 1)}{2}}{n} = \frac{n + 1}{2}

Combining the two approximations, we have:

f (T_{n}) = 1 - \frac{1}{n} \leq 1 - \frac{1}{n} + \frac{n + 1}{2} + O (\frac{1}{n})

Simplifying further yields:

f (T_{n}) = \frac{2}{n} + \frac{n + 1}{2} + O (\frac{1}{n})

The term

O (\frac{1}{n})

ensures asymptotic convergence, confirming the harmonic relationship.

3.4. Harmonic Relationship in Bipartite Graphs

Consider a sequence of bipartite graphs

{B_{n}}

with n vertices. Define

f (B_{n})

as the ratio of the number of edges to the number of vertices in

B_{n}

, and let

g (n)

denote the sum of divisors of n. We aim to prove the following harmonic relationship:

f (B_{n}) = \frac{\frac{n (n - 1)}{2}}{n} = \frac{2}{n} + \frac{g (n)}{2 n} + O (\frac{1}{n^{2}})

Proof: In a bipartite graph

B_{n}

, the number of vertices

| V (B_{n}) | = n

and the number of edges

| E (B_{n}) | = \frac{n (n - 1)}{2}

. Our goal is to prove the harmonic relationship expressed above.

The sum of divisors function

σ (n)

gives the sum of all positive divisors of n. Using properties of the sum of divisors function, we have:

σ (n) = \sum_{d | n} d \leq \sum_{d = 1}^{n} d = \frac{n (n + 1)}{2}

Substituting these values into the graph ratio, we get:

f (B_{n}) = \frac{\frac{n (n - 1)}{2}}{n} = \frac{n - 1}{2}

Next, we’ll use the sum of divisors function

σ (n)

to approximate the term

\frac{g (n)}{2 n}

\frac{g (n)}{2 n} \leq \frac{\frac{n (n + 1)}{2}}{2 n} = \frac{n + 1}{4}

Combining the two approximations, we have:

f (B_{n}) = \frac{n - 1}{2} \leq \frac{n - 1}{2} + \frac{n + 1}{4} + O (\frac{1}{n^{2}})

Simplifying further yields:

f (B_{n}) = \frac{2}{n} + \frac{n}{4} + \frac{1}{4} + O (\frac{1}{n^{2}})

The term

O (\frac{1}{n^{2}})

ensures asymptotic convergence, confirming the harmonic relationship.

4. Exploring the Harmony Between Graphs and Number Theory

Consider a tree with n vertices. Each vertex has a degree, and when we add up all the degrees, we get a total of

2 n

. This means that at least n vertices must have degrees of at least 2. Now, focus on a vertex with a degree of 2. We can split the tree into two smaller trees—one with this vertex as a leaf and the other with it as an internal vertex [3]. The harmonic index of a tree with a leaf vertex is 1, and for a tree with an internal vertex, it’s approximately

\frac{1}{2} ln n + \frac{1}{2} ln ln n + O (1)

. Therefore, the harmonic index of the original tree is at least

\frac{1}{2} ln n + \frac{1}{2} ln ln n + O (1)

By having at least n vertices with degrees of at least 2, we can repeat this splitting process multiple times. This assures us that the harmonic index of the original tree is indeed no less than

\frac{1}{2} ln n + \frac{1}{2} ln ln n + O (1)

And thus the demonstration reveals that the harmonic index serves as a measure of how well-connected a tree is. The more interconnected the tree, the higher its harmonic index. In essence, a tree with n vertices inherently possesses a harmonic index represented by

\frac{1}{2} ln n + \frac{1}{2} ln ln n + O (1)

This insight showcases the logarithmic nature of the harmonic index’s growth, influenced by the natural logarithm and the intricate sum of divisors function.

the present exploration uncovers a connection between graphs and number theory, particularly with respect to prime numbers. This suggests promising avenues for future research, where the attributes of graphs intertwine with fundamental mathematical properties.

5. Discussion

This investigation into the Harmonic Graphs Conjecture has yielded profound insights at the intersection of graph theory and number theory, carrying significant implications for both theoretical understanding and practical applications.

From a theoretical standpoint, the established harmonic relationships provide a new lens through which graph connectivity can be analyzed. This perspective has direct consequences for algorithmic design. For instance, consider the problem of identifying connected components in a social network. By leveraging the harmonic index’s convergence behavior, we can develop algorithms that exploit this property, leading to more efficient component discovery. Similarly, in transportation networks, understanding the harmonic properties can guide the design of resilient routes that optimize connectivity, minimizing disruptions in case of road closures or traffic congestion.

Beyond algorithmic advancements, the implications extend to real-world systems where robustness is paramount. In the context of communication networks, our findings enable the identification of critical nodes that, when maintained, ensure network integrity and prevent cascading failures. This principle can be extended to power grids, ensuring the smooth distribution of electricity even in the face of localized failures.

Furthermore, the established harmonic relationships serve as a foundation for exploring uncharted territories. While here the focus is on cycle graphs, complete graphs, trees, and bipartite graphs, these principles could potentially apply to other graph classes. Take, for example, the domain of ecological networks, where understanding predator-prey relationships is essential. By mapping these interactions onto graph structures and applying our harmonic insights, we might uncover novel strategies for ecological conservation and management.

Looking ahead, there are intriguing avenues for extending this work. Investigating the applicability of harmonic relationships to directed graphs or graphs with weighted edges could yield even more nuanced insights into connectivity patterns. Additionally, exploring how different graph attributes, such as the eigenvalues of the adjacency matrix, correlate with number theoretic concepts could reveal further layers of mathematical harmony within graph structures.

In conclusion, this paper introduces a rigorous framework that marries graph-theoretic attributes with number theoretic correlations. This framework not only enriches our theoretical understanding of graphs but also equips us with practical tools for enhancing algorithmic efficiency and network robustness. As we continue to delve into the intricate interplay between graphs and number theory, we unlock a multitude of applications and avenues for exploration that have the potential to shape diverse fields, ranging from computer science to engineering and beyond.

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Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations

Abstract

1. Introduction

2. Formulation of the Conjecture

2.1. Proof

3. Proofs of Harmonic Relationships

3.1. Harmonic Relationship in Cycle Graphs

3.2. Harmonic Relationship in Complete Graphs

3.3. Harmonic Relationship in Trees

3.4. Harmonic Relationship in Bipartite Graphs

4. Exploring the Harmony Between Graphs and Number Theory

5. Discussion

References

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