BALANCED UNITARY CAYLEY SIGRAPHS OVER FINITE COMMUTATIVE RINGS

Let R be a finite commutative ring with identity 1. The unitary Cayley graph of R, denoted by GR, is the graph whose vertex set is R and the edge set {{a, b} : a, b ∈ R and a - b ∈ R×}, where R× is the group of units of R. We define the unitary Cayley signed graph (or unitary Cayley sigraph in short) to be an ordered pair ????R = (GR, σ), where GR is the unitary Cayley graph over R with signature σ : E(GR) → {1, -1} given by [Formula: see text] In this paper, we give a criterion on R for SR to be balanced (every cycle in ????R is positive) and a criterion for its line graph L(????R) to be balanced. We characterize all finite commutative rings with the property that the marked sigraph ????R,μ is canonically consistent. Moreover, we give a characterization of all finite commutative rings where ????R, η(????R) and L(????R) are hyperenergetic balanced.