Line graph of extensions of the zero-divisor graph in commutative rings

Abstract

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 E G ( P ) with Z ( P ) * = Z ( P ) ∖ { 0 } as the vertex set, and two distinct vertices x and y are adjacent if and only if a n n ( x y ) is an essential ideal of P . At the same time, the weakly zero-divisor graph of P is denoted by W Γ ( 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 ∈ a n n ( u ) * and s ∈ a n n ( v ) * such that r s = 0 , where a n n ( u ) = { v ∈ P : u v = 0 } for u ∈ P . In our research, we deal with the conditions under which both E G ( P ) and W Γ ( P ) can be classified as line graphs. Furthermore, we explore the scenarios in which these graphs are the complements of line graphs.