Cayley graphs and cayley Signed graphs over finite commutative rings

Let R be a finite commutative ring with identity1≠0. The unitary Cayley graph of R, denoted by Gr, is the graph whose vertex set is R and the edge set {{a,b} : a,bϵR and a-bϵR ˟ }, where R ˟ the group of units of R. We study a unitary Cayley signed graph and characterize all finite commutative ring in which their signed graphs are balanced. We find the energy of a subgraph of unitary Cayley graph induced by square mapping. Moreover, we determine the spectrum and obtain the energy of a Cayley graph over a finite chain ring and apply them to get further results on a gcd-graph over a quotient ring of a unique factorization domain.