Signed Zero-Divisor Graph

Let R be a finite commutative ring with unity (1≠0) and let Z(R)⁎ be the set of non-zero zero-divisors of R. We associate a (simple) graph Γ(R) to R with vertices as elements of R and for distinct x,y∈R, the vertices x and y are adjacent if and only if xy = 0. Further, its signed zero-divisor graph is an ordered pair ΓΣ(R):=(Γ(R),σ), where for an edge ab, σ(ab) is '+' if a∈Z(R)⁎ or b∈Z(R)⁎ and '−' otherwise. This paper aims at gaining a deeper insight into signed zero-divisor graph by investigating properties like, balancing, clusterability, sign-compatibility and consistency.