Graph theory and algebra are two disciplines of mathematics which concentrate on building and investigating structures. Algebra is a fundamental branch of mathematics, whose roots are traced back to the early sixteenth century, whereas, graph theory is a flourishing mathematical research ground, which unfolded in the early eighteenth century, as the Swiss mathematician solved the famous Königsberg bridge problem, by representing the structure of the bridge and the landmass surrounding it as a graph. Hence, the subject emerged as a consequence of modeling real-life problems in terms of graphs, as it gives a comprehensive visual representation of the problem, and this aids in obtaining optimal and feasible solutions to the problem. It is interesting to note that, along with the increase in applications of the developed theories, the theory by itself has evolved independently over the period of time and has established itself as a flourishing mathematical discipline.

An algebraic structure is a non-empty set along with one or more operations (usually binary) defined on it and by the very definition of a graph, it can be noticed that a graph can be realised as a structural representation of a relation defined on a (vertex) set. Relating these two structural aspects, a synergy between the algebraic and graphical structures is studied in the field of algebraic graph theory. It has become a stimulating research field, yielding numerous intriguing results as these two disciplines; algebra and graph theory, interact in many ways to mutually extend the tools of one subject for the benefit of the other. In fact, powerful combinatorial methods found in graph theory have been used to prove specific significant and well-known results in group theory. For example, all finite groups can be represented as the automorphism group of a connected graph (c.f.[1]).

Any algebraic structure can be interpreted as a graph, and there are multiple ways of associating an algebraic structure with a graph. In the past few decades, several graphs are being constructed from algebraic structures based on different properties that the algebraic structures posses, and these algebraic graphs have been studied extensively in a motivation to understand the algebraic structure more clearly; thereby making this an enthralling area of research (c.f.[2,3,4]).

This association of an algebraic structure with a graph began in the end of the nineteenth century, when Arthur Cayley connected graph theory and group theory by introducing the Cayley graph of a group (c.f. [5]), which encoded the algebraic information of a group as a graphical structure. The Cayley graph for a group $\mathcal{G}$ is a graph with the vertex set as the elements of the group $\mathcal{G}$, which is invariant under the right translation by elements of $\mathcal{G}$. Cayley graphs are by far the most well-known graph associated with an algebraic structure. They have a massive yet, growing literature to an extent to convince that algebraic graph theory is only the study of Cayley graphs of finite groups (see [6,7,8,9,10,11]).

Another important class of algebraic graph construction is the construction of graphs from rings, as the study of graphs constructed from rings contributes to an interplay between the ring structure and the corresponding graph structure. One can sometimes translate the algebraic properties of the rings in terms of graph-theoretic properties and vice-versa, which can help in exploring some interesting results related to the graphs as well as the rings. Graphs defined on rings either have vertices as the set of elements of the ring or they are intersection graphs such that each vertex represent some subset of the ring, or some well-known sub-structure of the ring like ideals, subring, etc. and the edges are defined with respect to an algebraic condition on the elements of the vertex set.

The study on graph defined from rings began with the introduction of the zero-divisor graphs, which is one of the most well-studied graph defined on commutative rings that have massive and still augmenting literature (see [12,13,14]). Apart from the zero-divisor graphs, there are several other graphs such as the total graphs, annihilating graphs, comaximal graphs, unit graphs, Jacobson graphs, generalized total graphs, etc. They all have substantial and growing literature (c.f. [13,14,15,16,17,18,19]). A few decades back, algebraic graph theory was just a theory that did not apply to ordinary human activities, whereas it has now been successfully used in transmitting encrypted information with high security and privacy through public communication networks (c.f.[20]).

Though Cayley graphs were initially constructed on groups, the graph construction has been extended to rings as well. As rings possess several symmetric subsets like the set of all zero-divisors, units, idempotent, nilpotent elements, etc. many variants of Cayley graphs using these symmetric subsets of the rings were constructed and studied. This literature review intends to present an overview of these variants of Cayley graphs that are defined on rings. That is, the graphs defined such that their vertex sets are the ring elements and their adjacency relation is similar to the adjacency condition given in the Cayley graph, with respect to some symmetric subset of the ring.

It can be seen that there are many survey papers, review papers and books on graphs defined on rings (see [12,16,17]), but many of them cover only several well-studied graphs. Furthermore, review papers that focus on a particular property of the graph defined on rings can also be found in the literature (c.f. [21,22,23]), whereas there was no comprehensive review found on the variants of Cayley graphs defined on rings. This motivated us to create a literature hub on these graphs defined on common grounds, and systematically analyse the study that has been done on these graphs to understand the pattern and dynamics of research in this area. This systematic review also helps to identify unsolved open problems that were proposed in the literature as well as the future scope of study on the topic. Also, this article aims to clear the ambiguity over different graphs with similar names and the same graphs with different names that have been defined and studied independently by different authors, which falls under this criteria.

The outline of the article is as follows. The graph theoretic and algebraic preliminaries that are required to proceed further are given in Section 1.1. A comprehensive review on the unitary Cayley Graph of ${\mathbb{Z}}_n$ and unitary Cayley graph of a ring, where the former is a particular case of the latter is given in Section 2 and Section 3 respectively. This is followed by a review on the unitary addition Cayley graph in Section 4 and the unit graph of a ring in Section 5, where again the first class of graphs forms a subset of the second one. Finally, a review on other variants of Cayley graphs, for which detailed investigations are not yet done, is given in Section 6 and we conclude the article by proposing the research gaps that we have found over the course of the review along with several possible avenues for further research in Section 7.

One of the well-studied graphs defined on rings, especially on ${\mathbb{Z}}_n$, is the unitary Cayley graphs. As the name suggests, the unitary Cayley graphs can be seen as a restriction or a variation of the broadly defined Cayley graphs. As this graph is specifically defined on ${\mathbb{Z}}_n$, it can be seen that the number-theoretic definition of the graph leads to several interesting results that are obtained using number-theoretic properties and often the innate structure of the graph gives rise to pleasing combinatorial results.

A graph of order n is said to be representable modulo k if its vertices can be labeled using distinct integers between 0 and k such that the difference of the labels of two vertices are relatively prime to k if and only if the vertices are adjacent and the smallest k for which the graph is representable modulo k is called the representation number of the graph (see [33]). The problem of determining the representation number of a given graph and analysing the property of graphs that have a given representation number, along with its relation between the order of the graph was one of prominent that was put forth as the graph representation problem in the last decade of the twentieth century, as it was proved that every graph is representable modulo for some positive integer (c.f.[33]). The main motivation to study the unitary Cayley graph on ${\mathbb{Z}}_n$ was to investigate the representation problem of graphs, put forth in [33], which is closely related to the definition of the unitary Cayley graph on ${\mathbb{Z}}_n$ given below, following which an example of a unitary Cayley graph is given in Figure 1.

Note that the definition of the unitary Cayley graph of ${\mathbb{Z}}_n$ is closely associated with the definition of a graph to be representation modulo n and therefore, motivated to gain insights on the graph representation problem, the unitary Cayley graphs were investigated. It can be observed that post the introduction of the unitary Cayley graph $X_n$, the definition of a graph to be representable modulo n was given in terms of $X_n$. In other words, a graph is said to be representable modulo k if it is isomorphic to an induced subgraph of $X_n$ (refer to [35]).

Though the representation problem is stated in terms of the unitary Cayley graphs, $X_n$ and the results obtained on the investigation of the representation problem may be related to the graph $X_n$, note that we do not consider them in the review as the results may address only certain induced subgraph structures of the graph $X_n$, which may or may not have all the properties of $X_n$.

The unitary Cayley graph of ${\mathbb{Z}}_n$ was introduced in [34] as a specific case of the Cayley graphs defined using the generating sets of ${\mathbb{Z}}_n$, as the set ${\mathbb{Z}}_n^{*}$ generates ${\mathbb{Z}}_n$. The other variants of Cayley graphs defined based on generating sets in [36] were complete graphs and based on coloring the edges of these complete graphs in a symmetric fashion, the realisation of the induced subgraphs of these complete graphs as totally multicolored (TMC) subgraphs; that is, a subgraph of a graph in which no two edges have the same color, was studied in [36].

Motivated to investigate the possibilities of obtaining totally multicolored Cayley graphs, the unitary Cayley graph was defined on ${\mathbb{Z}}_n$ and its basic properties were investigated in [34]. By Definition 1, it can be seen that the graph $X_n$ is $\varphi \left(n\right)$-regular, where $\varphi \left(n\right)$ is the Euler’s totient function that gives the number of integers less than n that are relatively prime to n. The symmetric nature of the graph can be observed from the adjacency pattern as well as the regularity, as it is closely related to the number theoretic concepts of modular arithmetic (c.f. [37]). This symmetry of the unitary Cayley graphs gives raise to several applications in modelling networks and encourages the investigation on the graph in several directions.

The primary focus of the study in [34] was to examine the existence of triangles and the enumeration of them in the newly defined unitary Cayley graph, as the intended study was to explore the possibilities of obtaining totally multicolored graphs. This study on the triangles present in the graph helps to identify TMC graphs, but it can be seen that the study shall not be significant when the graph turns out to be a complete graph. Therefore, the first result obtained on $X_n$ classifies the values of n for which $X_n$ is a complete graph. Since bipartite graphs are characterised based on the existence of odd cycles, the values of n for which $X_n$ is bipartite and complete bipartite were also obtained as follows.

It can be observed that the graphs $X_{2^t}$, $t\in \mathbb{N}$ are regular, with each vertex having degree equal to half the number of vertices and this makes the size of the graph as the square of the sum of degrees of all vertices in the graph. Since the chromatic uniqueness of complete bipartite graphs was proved in [38], the graphs $X_{2^t}$, $t\in \mathbb{N}$ are called chromatically unique unitary Cayley graphs. Note that for a graph G, the polynomial that gives the number of graph colorings as a function of the number of colors is a chromatic polynomial (see [25]) and two graphs $G_1$ and $G_2$ are chromatically equivalent if they have the same chromatic polynomial; that is, $P_{\alpha }\left(G_1\right)=P_{\alpha }\left(G_2\right)$ and a graph $G_1$ is said to be chromatically unique if $P_{\alpha }\left(G_1\right)=P_{\alpha }\left(G_2\right)$ implies that $G_1\cong G_2$ (see [39]).

As the graph $X_n$ is triangle-free for even n, the enumeration of triangles were restricted to $X_n$, for odd n. As a first step, the number of triangles in $X_n$ with two common vertices were enumerated, following which the total number of triangles in the graph was determined. The number of triangles with two common vertices was obtained as the cardinality of the set $\{u\in {\mathbb{Z}}_n^{*}:(u-1)\in {\mathbb{Z}}_n^{*}\}$. This is because, the vertex set of any triangle in $X_n$ with two common vertices can be taken as $\{0,1,u:u\in {\mathbb{Z}}_n^{*}\}$, owing to the fact that the difference between the vertices of any edge in the graph is a unit. Therefore, the third vertex that differs for the triangles with two common vertices will always be a unit and hence, the number of triangles with two common vertices is obtained as where the product is runs over all the prime factors of n.

To enumerate the number of triangles in the graph $X_n$, the group action of the group, ${\mathbb{Z}}_n^{*}\times {\mathbb{Z}}_n$ on the set of all triangles of the graph; that is, if $(u^{\prime },x)\in {\mathbb{Z}}_n^{*}\times {\mathbb{Z}}_n$, then the action $(u^{\prime },x)\{0,1,u\}=\{u^{\prime }x,u^{\prime }(1+x),u^{\prime }(u+x)\}$ that gives the orbits of the triangles corresponding to different pairs $(u^{\prime },x)\in {\mathbb{Z}}_n^{*}\times {\mathbb{Z}}_n$ was considered. As orbits partition a set, the sum of the cardinalities of these orbits obtained through the given group action aided in determining the total number of triangles in the graph $X_n$. Using the orbits obtained through the group action, the edges of the triangles were also colored to obtain the edge coloring of the graph and this led to the enumeration of triangles having different possible combination of colors; that is, the triangles that have all three edges colored with different colors, all three edges colored with the same color and two edges colored with same color were termed as scalene-color triangles, equilateral-color triangles and isosceles-color triangles and they were enumerated.

The enumeration of triangles in the unitary Cayley graphs gave rise to the problem of counting the number of induced cycles of any given length k. Also, it was seen that to prove the chromatic uniqueness of a graph, it is important to count the number of induced k cycles in the graph, as some of the coefficients in the chromatic polynomials are related with the number of such induced cycles (see [40]). Therefore, this problem of counting the induced k cycles was proposed in [41] and the induced cycles of length 4 were enumerated using the concept of the multiplicative arithmetic property (map) of the graphs $X_n$.

A sequence of Cayley graphs $Cay({\gamma }_t,S_t)$, where ${\gamma }_t$ is an Abelian group and $S_t$ is a symmetric subset of ${\gamma }_t$, is said to have the multiplicative arithmetic property if for each pair of positive relatively prime integers $(n_1,n_2)$, there is a group isomorphism ${\varphi }_{n_1,n_2}$ from ${\gamma }_{n_1{n}_2}$ to ${\gamma }_{n_1}\times {\gamma }_{n_2}$ such that ${\varphi }_{n_1,n_2}$ maps $S_{n_1{n}_2}$ onto $S_{n_1}\times S_{n_2}$ (see [41]). In [41] the multiplicative arithmetic property on all the Cayley graphs defined on Abelian groups were discussed and since ${\mathbb{Z}}_n$ is also an Abelian group and ${\mathbb{Z}}_n^{*}$ is a symmetric subset of ${\mathbb{Z}}_n$, the unitary Cayley graphs were also examined in [41].

In [41], all Cayley graphs defined on Abelian groups were proved to have the multiplicative arithmetic property by obtaining the corresponding multiplicative arithmetic functions. A construction of sequences of Cayley graphs with the multiplicative arithmetic property, based on the number theoretic concepts like the Chinese reminder theorem was also given in the article. As an application of proving the multiplicative arithmetic property of the unitary Cayley graphs, the number of induced cycles of length 3 (triangles) and 4 were enumerated. Though, the formula for the number of triangles had been obtained previously in [34] using the group actions, the same result was deduced in this article using the multiplicative arithmetic property of the graph.

Along with the results obtained, the authors had also posted many open problems, among which the possibility to obtain a generalised expression to find the number of induced k cycles in the graph $X_n$, for any given n and to characterise the chromatic uniqueness in $X_n$ pertains to the unitary Cayley graphs. These open problems were partially addressed by the same authors in [42], by establishing a connection between the existence of an induced k cycle in $X_n$ and the number of prime divisors of n as follows.

This result was proved based on the results obtained in [41], that established the multiplicative arithmetic property of the unitary Cayley graphs. By Theorem 2, it was deduced that $X_n$ is a complete p-partite graph on n vertices with the maximum number of edges and is chromatically unique, when $n=p^t$, where p is prime and $t\in \mathbb{N}$, with the partitions $P_i=\{x:x\cong imodp,0\le i\le p-1\}$ . In [34], it was obtained that $X_n$ is chromatically unique when $n=2^t$, for some $t\in \mathbb{N}$ based on the structure of the graph, and this result is extends the class of chromatically unique unitary Cayley graphs from n being only $2^t$ to any prime power, $p^t$ and this result was also proved based on the multiplicative arithmetic property. Along with this, the bounds for the value $M\left(r\right)$ are also obtained as follows.

The bounds given in Theorem 3 shows the existence of induced k cycles in $X_n$, for arbitrarily large r, which adds credibility to Theorem 2. Also, a large gap between the bounds of $M\left(r\right)$ opened an avenue to find better estimates, which were computed in [43]. The main problem addressed in [43] was to determine the length of the longest induced cycle in $X_n$ for a given n and to address this problem, a representation of the vertices in $X_n$ based on their residues modulo the prime factors of n, called the residue representation is introduced as follows.

This representation simplifies the problem of finding the induced cycles in the graph to that of checking the similarity conditions between consecutive vertices; that is, to check if any pair of non-consecutive vertices has at least one same index in the representation, as it can be observed that for any $x,y\in V\left(X_n\right)$, $xy\in E\left(X_n\right)$ if and only if $x\equiv ymod{p}_i$ for all $1\le i\le r$. In this article, the number $M\left(r\right)$ defined in [42] is given in terms of $m\left(n\right)$, which denotes the longest induced cycle in $X_n$ as $M\left(r\right)={max}_n\left\{m\left(n\right)\right\}$, where the maximum is taken over all n values with r distinct prime divisors. Since $M\left(r\right)$ was proved to depend only on r in [42], $m\left(n\right)$ was also proved to depend only on r in [43], so that there arises no ambiguity in the given definition of $M\left(r\right)$ in terms of $m\left(n\right)$. Significant questions on the relation between the values $m\left(n\right)$ and $M\left(r\right)$ were also answered in [43], from the conditions under which these values of $m\left(n\right)$ and $M\left(r\right)$ are equal were obtained as given below.

Theorem 5 reduces the complexity of calculating $m\left(n\right)$ for large values of n, as it considers only the values of n whose prime powers are square-free. These results aided in improving the tightness of the bounds of $M\left(r\right)$ in [43], which is given below.

To prove Theorem 6, an induced subgraph of $X_n$ with $2^r+2$ vertices was constructed for all n, and it was proved that the construction depends only on the number of prime divisors, r of n and not on the value of the prime divisors, thus providing a lower bound for $m\left(n\right)$. It was natural to examine the properties of $X_n$ that contributed to the results that were obtained and to explore the possibilities of constructing similar graphs. On analysing these properties, it was noted that the above results on the length of the longest cycles can be extended to the direct product of any number of complete k-partite graphs and this extension can be seen as an immediate consequence of the fact that for any $n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}{p}_3^{{\alpha }_3}\dots p_r^{{\alpha }_r}$, $X_n\cong X_{p_1^{{\alpha }_1}}\times X_{p_2^{{\alpha }_2}}\times \dots \times X_{p_r^{{\alpha }_r}}$, as $X_n$ is a complete p-partite graph for $n=p^t$, when p is prime. Note that the unitary Cayley graphs are referred to as the unitary circulant graphs in [43].

A random walk on a finite, connected graph is a Markov chain1 that jumps from a current vertex v to one of its k neighbors, where with a uniform probability (refer to [45]). The hitting time $T_v$ of a vertex v is the minimum number of steps that a random walk takes to reach back the same vertex and the expected value of $T_v$ for a vertex is known as the expected hitting time. The expected hitting times for the random walks in the unitary Cayley graph $X_n$ and the direct product of two unitary Cayley graphs $X_{n_1}$ and $X_{n_2}$, where $n_1=p^{t_1}$ and $n_2=p^{t_2}$,$t_1,t_2\in \mathbb{N}$ were studied in [46] and [47] respectively, as an extension of the study on the expected hitting time of the edge transitive graphs by the same authors in [45]. Though the high symmetry of the graph $X_n$ can be realised from the graph construction, the unitary Cayley graphs were formally proven to be arc-transitive in [46], by obtaining an automorphism of the graph that satisfies the condition of arc transitivity as follows.

A graph G is said to be a vertex-transitive (edge transitive) graph if its automorphism group acts transitively on $V\left(G\right)$ ($E\left(G\right)$). In other words, a graph G is vertex-transitive (edge-transitive), if there exists an automorphism between any two distinct vertices (edges) of G. Similarly, a graph G is arc-transitive if there exists an automorphism between any two distinct edges of G such that the direction of the edges are preserved.

As it can be observed that arc-transitive graph is both vertex transitive and edge transitive, and hence, this automatically proves that the unitary Cayley graphs are both vertex and edge transitive. The main focus of the article [46] was to determine the expected hitting time of the edge transitive graphs, when the diameter of the graphs are 2 and 3, and to tighten the results when the graphs follow certain adjacency patterns. Since Theorem 7 proves the edge transitivity of the unitary Cayley graphs, the expected hitting times of these graphs were explicitly computed in [46] by classifying the graphs that have diameter 2 and 3 as follows.

By the definition of a random walk, it can be noted that the study of random walk in a regular graph tends to give a uniform distribution, as the number of neighbors to which the vertex can jump is equal for all the vertices in the graph. Also, the unitary Cayley graphs considered in the study were the graphs $X_n$, $n=p^k$, where p is a prime which were already proven to be complete k-partite graphs in [42]. To determine the hitting times of these graphs, the degree and distance between each pair of vertices in the graph must be known and therefore, the degree and distance between each pair of vertex in the graph $X_n$, when n is a prime power was determined in [46], and the diameter of the graph $X_{n_1}\times X_{n_2}$, where $n_1=p^{r_1}$ and $n_2=p^{r_2}$; for $r_1,r_2\in \mathbb{N}$, was also determined as 2 in [47]. As the graphs $X_{n_1}\times X_{n_2}$ are of diameter 2, the hitting time of the vertices of these graphs were also computed and are given as follows.

Though the unitary Cayley graphs were officially introduced in [34] in the year 1995, not many studies had emerged on the unitary Cayley graphs until 2007, before [48] was published. It was the first study that laid a strong foundation to the study on the unitary Cayley graphs, as it had an in-depth investigation on the properties of the unitary Cayley graphs; only after which, a huge growing literature can be found on the topic. The study in [48] begins with a brief review on the previous investigations of the unitary Cayley graphs, following which the chromatic number, clique number, and the vertex connectivity of $X_n$ were computed as follows.

An arc-transitive graph for which the vertex connectivity being its degree makes the unitary Cayley graphs highly reliable and stable for networks models. Also, the regularity of the graph $X_n$ implies that its complement is also regular and highly symmetric and therefore, using Theorem 10, the chromatic and clique number, $\chi \left({\overline{X}}_n\right)$ and $\omega \left({\overline{X}}_n\right)$ of the complement of $X_n$ were computed as $\frac{n}{p}$, where p is the smallest prime divisor of n. Based on these results on the complement of the unitary Cayley graphs, the following realisation was obtained.

Based on the investigation of the complement of the graph $X_n$ and its regularity, the number of common neighbors between the vertices were enumerated in [48] by partitioning the vertex based on different conditions for different values of n. On obtaining the chromatic and the clique number of the graphs, perfection in the unitary Cayley graphs was studied by investigating the existence of odd cycles of length 5 or more in the graph $X_n$ and the unitary Cayley graphs that are perfect were characterised as follows.

The investigation of the spectral properties of the unitary Cayley graphs began in [48], where the adjacency matrix of the graph $X_n$ was obtained. It is known that there are multiple adjacency matrices for any graph, which are given based on different ordering of the vertices. With the natural order of vertices $0,1,2,\dots ,n-1$, the adjacency matrix of the unitary Cayley graphs were obtained as circulant matrices; that is, matrices in which the entries of its first row generate the entries of the other rows by a cyclic shift, which established that the unitary Cayley graphs are circulant graphs; the graphs with circulant adjacency matrices (c.f. [20]).

Using the explicit formula to obtain the eigenvalues of a circulant matrix given in [49], the eigenvalues of the adjacency matrix of $X_n$ was obtained in terms of an arithmetic function $c(r,n)$ called the Ramanujan sum2, which takes only integral values for the given integers $r,n,n>0$. Therefore, it was concluded that all eigenvalues of unitary Cayley graphs are integers and hence, the unitary Cayley graphs fall under the class of graphs called the integral circulant graphs; circulant graphs whose eigenvalues are integers (see [50]). Further investigation on the eigenvalues of the graph $X_n$, based on their symmetry and the number theoretical properties had led to following interesting results on the eigenvalues of the graphs.

Fascinated by the spectral properties of the unitary Cayley graphs and its close relation with number theory, the authors defined a generalisation of the unitary Cayley graphs, called the $GCD$-graphs, in which the set of all positive, proper divisors of an integer $n>1$ is considered as the symmetric subset, to define the adjacency condition. The formal definition of the graph is given below.

Observe that the set $D_n^{*}$ consists of only all the proper positive divisors because when one is included as a divisor, the graph obtained shall be the complement of $X_n$, for certain values of n. The analysis on the spectra of $GCD$-graphs in [51] proved that the $GCD$-graphs also have integral eigenvalues. On further exploration of the properties of these graphs that have integral spectra, the authors came up with a slightly modified definition of the graphs based on this basic definition of the $GCD$-graphs that was put forth by them in [48], to obtain multiple smaller graphs which fall under this broad category with similar properties as follows.

Note that if $|D_n|=k$, then $2^{k-1}$gcd-graphs $G_n\left(D\right)$ are possible for any integer n,where the graphs $X_n$ and $X_n\left(D_n\right)$ are also one among them. An illustration of some gcd-graphs emerging from ${\mathbb{Z}}_{12}$, for the subsets $D\subset D_n$, apart from $D=\left\{1\right\}$ and $D=D_n$ is given in Figure 3.

This new generalised definition was simultaneously given in [51] in the process of characterising integral circulant graphs and it was proved that a graph is an integral circulant graph if and only if it can be realised as the graph $G_n\left(D\right)$, for some $D\subseteq D_n$. It can be observed that when the set of all proper divisors are considered, the gcd-graphs $G_n$ will be the $GCD$-graph defined in [48] and when $D=\left\{1\right\}$, $G_n\left(1\right)=X_n$.

Therefore, it can be seen that the unitary Cayley graphs can be realised as a special case of $GCD$-graphs as well as the gcd-graphs from their definitions, and any study on gcd-graphs can be considered to obtain results on the the unitary Cayley graphs. Also, based on the characterisation of the integral circulant graphs as gcd-graphs and the fact $G_n\left(1\right)=X_n$, the results established for the integral circulant graphs will also hold for the unitary Cayley graphs. The integral circulant graphs or the graphs $G_n\left(D\right)$ have a huge, growing literature, owing to its spectral properties that have applications in fields like chemistry, quantum physics, radiology, etc (c.f. [50]).

As already seen, the unitary Cayley graph $X_n$ is a special case of the integral circulant or gcd-graphs and hence, all the properties that are investigated for the latter shall hold for $X_n$, but the bounds and results obtained for the unitary Cayley graphs shall be more specific and tight than results obtained for these broader classes of graphs. Therefore, in this article, we present a review of the study which aree specifically made on the Unitary Cayley graphs and the results that were explicitly stated for the graph $X_n$, as an application or a corollary in the articles that study the integral circulant graphs or gcd-graphs.

In [48], an open problem to determine the automorphism group of the unitary Cayley graphs $X_n$, for $n>6$ had been posted by the authors, which led to the investigation on the automorphisms of $X_n$. Though the problem was not fully addressed, a necessary and sufficient condition for a bijective mapping to possess the structure of an automorphism of the graph $X_n$ was given in [53] as follows.

Apart from the above mentioned result that was obtained on the automorphism of $X_n$, a characterisation of planar unitary Cayley graphs was obtained along with the crossing number (The least number of edges that cross in a planar graph drawing.) of $X_n$ for few values of n for which the graph structure is a well-known graph class, using the existing results on the crossing number of these graph classes. The traversal properties of $X_n$ were also discussed in the article along with which the edge chromatic number and the edge connectivity of the graph were also determined as given below, where $\varphi \left(n\right)$ denotes the Euler’s totient function.

The property of the graph $X_n$ having both its edge and vertex connectivity equal to its degree of regularity and the graph being integral circulant, increases the application of the graphs in the field of networks, especially in areas that require a stable and strong network. This increases the significance of the study on the graph for various purposes and this also gives the researchers the curiosity to investigate other properties of the graphs, and construct similar graphs. Extending the study further, the authors studied the basic graph properties of the unitary Cayley graph of a ring, which is obtained as a finite direct product of the rings ${\mathbb{Z}}_n$, for different values of n. This extension gave rise to the idea of generalising the unitary Cayley graphs of ${\mathbb{Z}}_n$ to any ring R, a detailed review of which is given in Section 3.

The open problem to determine the automorphism group of $X_n$, put forth in [48] was solved in [54] by obtaining the automorphism groups of $X_n$ and their cardinality, for different values of n, as a tool to generalise the automorphism groups of the integral circulant graphs. The results obtained are given below and it shows that the structure of the automorphism groups are highly sophisticated as the value of n increases.

The structure of the automorphism group of $X_n$ was proved by partitioning the vertices of $X_n$ based on the residue modulo primes, which is similar to the residue representation introduced in [43] and the permutations on these residue classes were considered to obtain automorphisms of the graph, using the notion of modular arithmetic and the Chinese remainder theorem. According to the construction of automorphisms of $X_n$ in the proof of Theorem 22, it was concluded that the automorphism group is isomorphic to the wreath product of the permutation group (refer to [55]) of the graphs of residue classes modulo r and the permutation groups of vertices in each class, as given below.

The same problem of determining the automorphism group of the unitary Cayley graph was solved in [56,57], using different approaches. The study in [56] began with a motive to investigate the automorphism group of $X_n$; but the authors on observing the symmetric pattern of $X_n$ in several aspects, extended the concept of unitary Cayley graphs to any ring R and the automorphism groups of the unitary Cayley graphs defined on a ring R were investigated, which on special case of $R={\mathbb{Z}}_n$ gave the automorphism group of $X_n$. The main idea of their algebraic proof, where the dependence of the automorphisms on the underlying algebraic structure of the rings concerned was emphasized, is different from the proof given in [54], which used a number-theoretical approach. The authors of [57] investigated the automorphism group of the rational circulant graphs; circulant graphs with a rational spectra, in which the integral circulant graphs become a subclass, by developing a framework based on Schur rings (For more details, refer to [58,59]). The approach is highly complex as it is built for all rational circulant graphs; but it is claimed in [57] that the automorphism group of $X_n$ could have been traced a few decades ago if the framework of the approach presented in [57] was followed.

The results on the spectra of the unitary Cayley graphs obtained in [48] fascinated the researchers to explore other parameters and properties of the unitary Cayley graph $X_n$ that are closely associated with its adjacency matrix and its eigenvalues. The first of such properties to be investigated was the perfect state transfer in the unitary Cayley graphs. For a graph G with the adjacency matrix A, $H\left(t\right)$ is defined as the operator $e^{\left(itA\right)}$, called the transition operator. A perfect state transfer between the vertices u and v is said to happen at time $\tau$ if the $uv$-entry of ${|H\left(\tau \right)}_{u,v}|=1$. This perfect state transfer is being used in several areas that deals with allocation and assignment factors, especially it has been efficiently applied to key distribution in commercial cryptosystems, and in assignment of objects in quantum spin networks (see [50]). This notion was introduced to circulant graphs in [60] and the perfect state transfer in the integral circulant graphs was studied in [50]. Based on these studies, the class of unitary Cayley graphs that allow perfect state transfer was characterised in [50] as follows.

Following the study on perfect state transfer in the unitary Cayley graph $X_n$ the properties related to the energy of the graph, which is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph was determined in [61] and [62] as follows.

Theorem 26 arises as a consequence of Theorem 25 along with the fact that for $n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}{p}_3^{{\alpha }_3}\dots p_r^{{\alpha }_r}$ and $n^{\prime }=p_1{p}_2{p}_3\dots p_r$, where $p_i$, $1\le i\le r$ are distinct primes and $r\ne 1$, $X_n\cong X_{p_1^{{\alpha }_1}}\times X_{p_2^{{\alpha }_2}}\times \dots \times X_{p_r^{{\alpha }_r}}$. Based on the energy of the graph $X_n$ obtained, the hyperenergetic unitary Cayley graphs along with their complements were characterised in [61,62] as follows. Note that a graph G of order n is called hyperenergetic if its energy, $\mathcal{E}\left(G\right)$ is greater than the energy of the complete graph of order n; that is, $\mathcal{E}\left(G\right)>\mathcal{E}\left(K_n\right)=2(n-1)$ (see [61]).

Both [61] and [62] discuss the energy and hyperenergercity of the graphs $X_n$ and ${\overline{X}}_n$ and the same results using similar proof techniques were obtained independently. In addition to these results, the ratio $\frac{\mathcal{E}\left(X_n\right)}{2(n-1)}$ that measures the degree of hyperenergecity of $X_n$, which can be seen to grow exponentially as the number of distinct prime divisors of n increases, was given in [62]. In the process of proving the above results, the nullity of the graph was discussed, which was also independently proven in [63]. After the publication of [62], a comment on the article was released, wherein a one line proof to determine the energy of the unitary Cayley graphs that was determined in Theorem 25 and Theorem 26, using the notion of Ramanujan sums was given.

This was followed by a discussion on the eigenspace of the Unitary Cayley graphs in [64], where a specific case in the class of graphs called the Hamming graphs were proved to be isomorphic to the unitary Cayley graphs and using the results obtained on the spectra of these unitary Cayley graphs, the eigenspace of Hamming graphs were determined. Note that for non-negative integers $k,r,s$, the hamming graph $HG(l_1,l_2,...l_r;s)$ is a graph which is constructed based on the number of words formed by considering r out of a given k letters, which have a hamming distance s. In other words, given k letters, the $k^r$ possible words with $r\le k$ letters are the vertices of a hamming graph and two vertices are joined by an edge if their associated words differ in exactly s positions (see [64]).

A k-regular graph G is said to be a Ramanujan graph if and only if the second largest absolute value of the eigenvalues of the adjacency matrix of G, ${\lambda }_2\left(G\right)\ge 2\sqrt{k-1}$ (c.f. [65]). This idea of realising a graph as a Ramanujan graph was explored in unitary Cayley graphs and its complement, using the spectra of the graphs that were obtained in the previous literature and a complete characterisation of the cases in which the unitary Cayley graph and its complement are Ramanujan graphs was obtained in [65] and [66] respectively as follows.

Further investigation on some variants of energy, namely the distance energy, color energy, minimum covering Gutman energy, the minimum edge dominating energy and the Seidal Laplacian energy of the unitary Cayley graphs was conducted in [67,68,69,70,71,72] respectively. As already known, energy of a graph is the sum of the absolute values of the eigenvalues of a matrix. Based on the matrix defined, the corresponding spectra and the energies are computed. Therefore, the distance energy is obtained from the distance matrix of the graph, which is a square matrix in which the $ij$-th entry gives the shortest distance between the vertices $v_i$ and $v_j$ in the graph (see [69]). The color energy of a graph G corresponds to the energy of the $A_L$-matrix of G (c.f.[67,71]), whose entries are based on a proper vertex coloring of the graph G, say c, such that

A emphminimum covering set $C\subseteq V\left(G\right)$ of a graph G is a subset of vertices such that each edge of the graph is incident to at least one vertex in the subset, and the minimum number of vertices in such a set is called the minimum covering number of the graph (c.f.[70]). A minimum covering matrix $M{C}_C\left(G\right)$ of a graph G of order n is a $n\times n$ matrix defined based on the adjacency of the vertices in a minimum covering set C such that the diagonal entries of the adjacency matrix of the graph G is 1 if the corresponding vertex belongs to the minimum covering set considered (see [73]). The Gutman matrix $GM\left(G\right)$ of a graph G of order n is a square matrix of order n, whose entries are 0 and $d_i{d}_j{d}_{ij}$, where $d_i$ and $d_j$ are the degrees of the vertices $v_i$ and $v_j$ and $d_{ij}$ is the shortest distance between $v_i$ and $v_j$; corresponding to the conditions if the vertices $v_i=v_j$ and $v_i\ne v_j$ (c.f. [74]).

The minimum covering Gutman energy of a graph G is computed based on the minimum covering Gutman matrix $MCG\left(G\right)$ defined in [70], which as observed is defined as a combination of the minimum covering matrix and Gutman matrix as follows.

Similarly, the minimum edge dominating energy of a graph G is the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of G, which is a binary matrix of order $m\times m$, where m is the size of G in which the entries are based on the adjacency of the edges and the minimum edge dominating set of the graph. A subset $F\subseteq E\left(G\right)$ is an edge dominating set of a graph G if every edge not in F is adjacent to at least one edge in F and an edge dominating set with the least cardinality is called a minimum edge dominating set of the graph and cardinality is the edge domination number of the graph (c.f. [29]).

The study on minimum covering Gutman energy of $X_n$ involved the discussion of this energy for unitary Cayley graph $X_n$, for the values of n for which $X_n$ is a common graph class such as complete graph, complete multipartite graph, etc. A similar situation was encountered on the discussion of the minimum edge dominating energy of the unitary Cayley graphs in [68], except for a few bounds that were deduced instead of the exact values.

The distance spectra along with the corresponding energy of the unitary Cayley graphs was computed in [69], as a part of the study of the same on the integral circulant graphs and it was proved that the integral circulant graphs, including $X_n$, have integral distance spectra. On investigating the distance energies of both these graphs, a construction of infinite families of distance equi-energetic graphs (graphs, possibly isomorphic, that have the same energy) emerged, which were the first ones to be derived without using construction methods by taking graph products nor iterated line graphs (defined in the later part of this section). The results on the distance energy of $X_n$ and the construction obtained in [69] are given below.

In Theorem 34, the value of $|2\varphi \left(n\right)-2-\frac{n}{2}|$ cannot be resolved, since it takes all positive, zero and negative values and on specific n values, the solution of the problem relates to the still open conjecture on the Euler’s totient function (refer to [75]), for which obvious solutions involve prime Fermat numbers4.

The color energy of the unitary Cayley graph and its complement was studied in [67,71]. The eigenvalues of the $A_L$ matrix defined with respect to the proper colorings of the graphs were examined and the corresponding energy was obtained in terms of the chromatic number of the graph and the Euler’s totient function, using the notion of Ramanujan Sums. A study on a few other matrices of the unitary Cayley graphs along with their eigenvalues and energy was conducted in [76], where a small-world network depending on the unitary Cayley graph was proposed with an intent to decrease the delay and increase the reliability in data transfer and used to create and analyse network communication.

The Seidal Laplacian energy of the unitary Cayley graph $X_n$ was computed in [72] by obtaining the eigenvalues of the Seidal Laplacian matrix $SL\left(X_n\right)=S\left(X_n\right)-DS\left(X_n\right)$ of $X_n$, where $SL\left(X_n\right)$ is the Seidal Laplacian matrix of $X_n$, $S\left(X_n\right)$ is the Seidal matrix of $X_n$ and $DS\left(X_n\right)$ is an $n\times n$ diagonal matrix of $X_n$ which has its diagonal entries $n-1-2deg\left(v_i\right)$, $1\le i\le n$. The Seidal matrix of a graph G is an $n\times n$ matrix with entries 1, -1 corresponding to whether the vertices $v_i{v}_j\in E\left(G\right)$ or $v_i{v}_j\notin E\left(G\right)$ or 0, otherwise (refer to [72]).

An algebra over a field is an algebraic structure consisting of a set together with the operations of addition, multiplication and scalar multiplication by elements of a field that satisfies the axioms of a vector space with a bilinear operator5; that is, an algebra over a field is a vector space equipped with a bilinear operator (c.f. [77]). For a positive integer n, the set of all $n\times n$ matrices over the field of complex numbers, $\mathbb{C}$ forms an algebra ${\mathbb{M}}_n\left(\mathbb{C}\right)$, with the usual matrix multiplication. As the adjacency matrix of a graph $A\left(G\right)$ is a well-known square matrix, the adjacency algebra of a graph is defined as the subalgebra of ${\mathbb{M}}_n\left(\mathbb{C}\right)$ which consists of all polynomials of $A\left(G\right)$ with coefficients from $\mathbb{C}$, where a subalgebra is a subset of the algebra which is an algebra by itself under the same bilinear operator (refer to [78]).

The adjacency algebra of the unitary Cayley graph $X_n$ was investigated in [77]. Since every element of the adjacency algebra of a graph is a linear combination of the powers of its adjacency matrix, the results on the adjacency algebra of a graph was obtained using the powers of the adjacency matrix. Therefore, using the existing results in on the adjacency matrix of the graph $X_n$, the adjacency algebra of $X_n$ was discussed in [77] and it was proved that the adjacency algebra of unitary Cayley graphs is a coherent algebra; that is, it is a subalgebra of ${\mathbb{M}}_n\left(\mathbb{C}\right)$ containing $I,J$, where I is the identity matrix and J is the matrix with all its entries 1, which is closed under Hadamard product6 and conjugate transposition.

For a graph G with an adjacency matrix $A\left(G\right)$, its coherent closure, denoted by $\mathcal{CC}\left(G\right)$, is the smallest coherent algebra containing $A\left(G\right)$, and a graph G is said to be a pattern polynomial graph if its adjacency algebra is its coherent closure. On proving that the unitary Cayley graphs have a coherent adjacency algebra, the authors proved that every unitary Cayley graph is a pattern polynomial graph and using this, certain properties of the unitary Cayley graphs were derived based on the properties of pattern polynomial graphs, obtained in [79]. To prove that all unitary Cayley graphs are pattern polynomial graphs, the following characterisations on the structure of the graphs were obtained.

Recall that a k-regular graph G of order n is strongly regular with parameters $(n,k,r,s)$ if any two adjacent vertices have exactly rcommon neighbours and any two non-adjacent vertices have exactly s common neighbours and a crown graph, $C_{r,r}$ is a bipartite graph with vertex set such that $V\left(C_{r,r}\right)=V_1\cup V_2$ and $|V_1|=|V_2|=r$, with $V_1=\{v_1,v_2,\dots ,v_r\}$ and $V_2=\{u_1,u_2,\dots ,u_r\}$ such that $v_i{u}_j\in E\left(C_{r,r}\right)$ if and only if $i\ne j$.

Appropriate representation of the circulant graphs on a Euclidean plane, unveils the rotational symmetry of the graph. As known earlier, unitary Cayley graphs are integral circulant graphs and therefore such a suitable representation or drawing called the unit circle drawing of the unitary Cayley graphs were examined in [80]. The unit circle drawing of the graph $X_n$ is nothing but drawing the graph $X_n$ such that the vertices are placed equi-distantly on a unit circle on the complex plane $\mathbb{C}$ and the edges are drawn as line segments. This representation gives a hole like structure in the middle of the graph, which is called the central hole or the geometric kernel of the graph. Just like how the spectrum of a graph provides vital information on the graph, the size of the geometric kernel in the unit circle drawing of an integral circulant graph, which is measured through the kernel radius also provides the arithmetic properties of the graph.

It was proven in [81] that the central hole in the unit circle drawing of any circulant graph on $n>3$ vertices is a regular n-gon. Therefore, only the size of the geometric kernel for $X_n$, which is already known to be an n-gon had to be determined in [80], by computing the kernel radius, given by the formula $max\{k:1\le k<\frac{n}{2},gcd(k,n)=1\}$. Only integers less than $\frac{n}{2}$ are considered because there shall be no central hole when the edge $(k,\frac{k}{2})$ exists in the unit circle drawing of a graph. It was observed that the kernel radius of $X_n$ is a strictly decreasing function in the range $\left(0,\frac{n}{2}\right]$.

Apart from this, computation of certain graph parameters of the unitary Cayley graph were carried out in [82,83,84,85,86,87,88,89], where certain topological indices of the unitary Cayley graphs were computed in [86,87,88] and few graph polynomials for the unitary Cayley graphs were determined in [82], using the results that were given in [48], as graph polynomials are also graph invariants that codes numerical information about the underlying graph (c.f [90]).

It was already seen that the unitary Cayley graphs are highly reliable networks and can be used in modeling situations which require stable networks. To assert this and to study the degree of reliability of these networks, few vulnerability parameters which measures the vulnerability of a graph were computed for the unitary Cayley graphs in [84] and this study on computing vulnerability parameters paved way to examine the parameters related to graph covering in [89] and [91].

Graph covering problem is one of the most classical topics in graph theory, where the minimum number of the entities of a graph, like vertices, edges, etc. with a particular property having a given graph as their union is determined. One such covering parameter is the tree covering number, which is defined as the minimum cardinality among all tree covers of the graph, where a family of mutually edge disjoint trees in a graph is called a tree cover of the graph if each edge is an edge of a tree in the family. This tree covering number was determined for the unitary Cayley graph $X_n$ and its complement ${\overline{X}}_n$ in [89], from which the Nordhaus-Gaddum type inequalities; that is, bounds on the sum and the product of the invariant for a graph and the its complement, for the tree covering number were obtained. The exact value of the tree covering number of $X_n$ was computed as given in Theorem 38, whereas for the complement ${\overline{X}}_n$ the bounds according to different values of n were obtained. Based on these bounds, the Nordhaus-Gaddum type inequalities were also obtained for different cases of n depending on its prime factorisation.

The other aspect related to covering that was discussed for the unitary Cayley graphs in [91] was the property of the well-coveredness of a graph. A graph G is said to be well-covered if all its maximal independent sets are of the same size. In [91], the well-coveredness of the graphs $X_n$ and ${\overline{X}}_n$, along with its vertex decomposability were examined and the condition under which the graphs are well-covered and vertex decomposable (refer to [92] for more details on vertex decomposable graphs) were given. The number of walks between any pair of two vertices in the unitary Cayley graphs was enumerated in [83] and as an application of this result, it was shown that there exists a bijection between walks in $X_n$ and the ordered sums of units in ${\mathbb{Z}}_n$, using which the number of representations of a fixed residue class mod n as the sum of k units in ${\mathbb{Z}}_n$ was determined.

A function which is defined on the set of positive integers to a subset of the set of complex numbers is an arithmetic function. An arithmetic function h is multiplicative, if it is not identically zero, and for any $r,s\in \mathbb{N},h\left(rs\right)=h\left(r\right)h\left(s\right)$ whenever $gcd(r,s)=1$. For each non-negative integer r and prime p, the r-th Schemmel’s totient function $S{T}_r$ is a multiplicative arithmetic function that satisfies where $\alpha$ is a positive integer. From the name Schemmel’s totient function, it can be seen that this function introduced by Schemmel, is a generalisation of the Euler’s totient function $\varphi \left(n\right)$ (c.f.[93]). It can be seen that $S{T}_0\left(n\right)=n$ and $S{T}_1\left(n\right)=\varphi \left(n\right)$, for all integers n. Since most of the graph invariants of the unitary Cayley graph $X_n$ are computed and expressed in terms of $\varphi \left(n\right)$ and $S{T}_r\left(n\right)$ being its generalisation, it opened an avenue to check the possibility of expressing the parameters in terms of $S{T}_r\left(n\right)$ and in [94], a simple formula for the number of cliques of any order in the unitary Cayley graph $X_n$ was obtained as follows.

This formula naturally gives the number of triangles in the graph $X_n$ in terms of the Schemmel totient function as $\frac{S{T}_0\left(n\right)}{1}\frac{S{T}_1\left(n\right)}{2}\frac{S{T}_2\left(n\right)}{3}$, which is more generalised and simple than the same expression which was computed independently in [34,41,42,48].

The k-th power $G^{\left(k\right)}$ of a graphG is a graph whose vertex set is the same as the vertex set of G and there is an edge between two vertices in the graph $G^{\left(k\right)}$ if and only if there is a path of length at most k between them in G. The k-th power of the unitary Cayley graphs were examined in [85], where the energies of these graphs were determined and all the powers of unitary Cayley graphs that are Ramanujan graphs were classified. Note that in [85], the k-th powers of a unitary Cayley graph is addressed as the the distance powers of the graph. Using the results obtained on the energies of distance powers of unitary Cayley graphs, infinitely many pairs of non-cospectral equi-energetic graphs were constructed and all the hyperenergetic distance powers of unitary Cayley graph $X_n$ were characterised. It can be noticed that the k-th power of any graph G can be defined for the values $1\le k\le diam\left(G\right)$ and $diam\left(X_n\right)\le 3$. Therefore, the investigation is limited to the unitary Cayley graphs that have diameter 3, in which case there exists only the value $k=2$ for which the discussion of the k-th power of the graph $X_n$ is non-trivial.

Apart from Cayley graphs, the power graphs of groups have a growing literature, giving rise to several survey papers (c.f.[2,95,96,97]). Note that the power graph of a finite group is a graph with the vertex set as the elements of the group, and two vertices are adjacent if one is a power of the other and are not to be confused with the k-th power of a graph, as both the graphs are referred to as the power graphs in the literature. Owing to the huge literature on power graphs of finite groups, an open problem to explore the relation between the power graphs and Cayley graphs was put forth in [95]. This problem was addressed in [98] and it was shown that, for certain values of n, the vertex deleted subgraphs of power graphs of ${\mathbb{Z}}_n$ are spanning subgraphs or the complement of the vertex deleted subgraphs of certain unitary Cayley graphs. Using these relations, the relation between the energy of power graphs and Cayley graphs were also obtained in [98]. The following theorem gives a relation between the power graph $\mathcal{P}\left({\mathbb{Z}}_n\right)$ and unitary Cayley graph $X_n$ of ${\mathbb{Z}}_n$, for some values of n.