We present some observations on a restricted variant of unitary Cayley graphs modulo n, and implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs Gn, whose vertex set is the ring Zn, and where residues a, b modulo n are adjacent if and only if their difference is a quadratic residue. By bounding the diameter of...

Given an integer n, one defines the unitary Cayley graph, denoted Cay(Zn,Zn), to be the graph whose vertex set is Zn, the integers modulo n, with an edge between two vertices x, y if x− y is a unit in (the ring) Zn. Unitary Cayley graphs have been studied as objects of independent interest (see, for example, [3], [2], [7], [8], [9]) but are of particular relevance in the study of graph represen...

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Let G be a vertex colored graph. The minimum number χ(G) of colors needed for coloring of a graph G is called the chromatic number. Recently, Adiga et al. [1] have introduced the concept of color energy of a graph Ec(G) and computed the color energy of few families of graphs with χ(G) colors. In this paper we derive explicit formulas for the color energies of the unitary Cayley graph Xn, the co...

We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for any abelian group $G$ symmetric subset $S$. then focus on family of unitary $G_R=X(R,R^*)$, where $R$ is a finite commutative ring with identity. show under mild conditions, $\{G_R, G_R^+\}$ pairs integral non-isospectral (generically connected non-bipartite). Then, we obtain conditions such \bar G_R\}$ graphs. Finall...

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...