Unitary Addition Cayley Signed Graphs

Abstract

A signed graph (or sigraph in short) is an ordered pair S = (Su,), where Su is a graph G = (V, E) and : E !{+,−} is a function from the edge set E of Su into the set {+,−}. For a positive integer n, the unitary addition Cayley graph Gn is the graph whose vertex set is Zn, the ring of integersmodulo n and if Un denotes set of all units of the ring, then two vertices a and b are adjacent if and only if a + b 2 Un. For a positive integer n, the unitary addition Cayley sigraph n = (un,) is definedas the sigraph, where u n is the unitary addition Cayley graph and for an edge ab of n, phi(ab) =¨+ if a 2 Un or b 2 Un,− otherwise. In this paper, we have obtained a characterization of balanced and clusterable unitary addition Cayley sigraphs. Further, we have established a characterization of canonically consistent unitary additionCayley sigraphs n, where n has at most two distinct odd prime factors.