Properties of derived signed graphs on Beck’s zero-divisor graph

Abstract

The concept of associating a graph to a ring was initiated by Beck in 1988. The zero-divisor graph of a commutative ring [Formula: see text] with unity ([Formula: see text]) is defined as a simple graph, denoted by [Formula: see text], where elements of the ring [Formula: see text] represent vertices and two distinct vertices in [Formula: see text] are adjacent if the product of the corresponding elements is zero in [Formula: see text]. Here, we extend the notion of zero-divisor graphs to signed graphs. We introduce four types of signing to the zero-divisor graphs. Further, we characterize rings for which these four types of signed graphs, its lined signed graphs and their negations are balanced, clusterable, sign-compatible, canonically sign-compatible, and canonically consistent.