On the Unitary Cayley Signed Graphs

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E) and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a positive integer n > 1, the unitary Cayley graph Xn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a− b ∈ Un. For a positive integer n > 1, the unitary Cayley sigraph Sn = (Su n , σ) is defined as the sigraph, where Su n is the unitary Cayley graph and for an edge ab of Sn, σ(ab) = { + if a ∈ Un or b ∈ Un, − otherwise. In this paper, we have obtained a characterization of balanced unitary Cayley sigraphs. Further, we have established a characterization of canonically consistent unitary Cayley sigraphs Sn, where n has at most two distinct odd prime factors.