Line graph of extensions of the zero-divisor graph in commutative rings - Rehman, Nadeem ur; Mir, Shabir Ahmad; Nazim, Mohd (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 512.6 We consider a finite commutative ring with unity denoted by P. Within this framework, the essential graph of P is represented as EG(P) with Z(P)^* = Z(P){0} as the vertex set, and two distinct vertices x and y are adjacent if and only if ann(xy) is an essential ideal of P. At the same time, the weakly zero-divisor graph of P is denoted by WΓ(P) with Z(P)^* = Z(P){0} as the vertex set and an edge is defined between two distinct vertices u and v if and only if there exist r ann(u)^* and sann(v)^* such that rs=0, where ann(u) = {v Puv = 0} for u P. In our research, we deal with the conditions under which both EG(P) and WΓ(P) can be classified as line graphs. Furthermore, we explore the scenarios in which these graphs are the complements of line graphs.