一类有限交换环上的广义单位Cayley图的若干性质

Some Properties of a Generalization of the Unitary Cayley Graphs of a Finite Commutative Ring

设R是一个含有非零单位元的有限交换环,U(R)是R的单位群,G是U(R)的一个乘法子群,S是G的一个非空子集并且S-1={s-1|s∈S}S。单位Cayley图Cay(R,U(R))的顶点集是R,两个顶点x和y相邻当且仅当x-y∈U(R);而广义单位Cayley图Γ(R,G,S)的顶点集为R,两个顶点x与y相邻当且仅当存在s∈S,使得x+sy∈G。容易看出,当G=U(R)时,Γ(R,G,{-1})即为单位Cayley图。本文主要利用有限交换环的结构以及群与图的理论,研究了有限交换环上的广义单位Cayley图的一些性质,讨论了Γ(R,G,{s})的正则性,以及Γ(R,U(R),{s})中任意两点的公共邻接点个数和边着色数。

Let R be a finite commutative ring with non-zero identity and U（R） be the unit group of R. Suppose that G is a multiplicative subgroup of U（R）, and S is a non-empty subset of G such that S 1= {s^-1｜s∈S}∈S. Then the vertex set of unitary Cayley graph Cay（R,U（R）） is R, and two distinct vertices x and y are adjacent if and only if x-y∈U（R）. Γ（R,G,S） is a graph whose vertex set is R, and vertices x and y are adjacent if and only if there exists sES such that x＋sy∈G. Obviously, if G=U（R）, then Γ（R,G,{-1}） is the unitary Cayley graph. By the structure of a finite commutative ring and the theory of group and graph, we study some properties of a generalization of the unitary Cayley graphs of a finite commutative ring. We consider the regularity of a generalization of the unitary Cayley graphs Γ（R,G, {s} ）, and determine the number of common neighbors of two distinct vertices of Γ（R,U（R）, { s} ）. In addiction, evaluate the edge chromatic number of Γ（R,U（R）, { s}）.