Beck’s Zero-Divisor Graph in the Realm of Signed Graph

Abstract

For a commutative ring R with unity ($$1\ne 0$$), the zero-divisor graph of R, denoted by $$\varGamma (R)$$, is a simple graph with vertices as elements of R and two distinct vertices are adjacent whenever the product of the vertices is zero. Further, its signed zero-divisor graph is an ordered pair $$\varGamma _{\varSigma }(R):= (\varGamma (R), \sigma )$$, where for an edge xy, $$\sigma (xy)$$ is ‘$$+$$’ if either x or y or both is a nonzero zero-divisor and ‘−’ otherwise. This article aims at gaining a deeper insight into signed zero-divisor graphs by investigating properties such as balancing, clusterability, sign-compatibility, consistency, $$\mathcal {C}$$-sign-compatibility, and $$\mathcal {C}$$-consistency.