On The Line Graph of Graph with Respect to Idempotents of a Ring

Abstract

Let [Formula: see text] be a ring with nonzero identity and [Formula: see text] denotes the set of idempotents in [Formula: see text]. A graph of [Formula: see text] with respect to idempotents [Formula: see text] is a graph whose vertices are elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Line graph [Formula: see text] of a graph [Formula: see text] is defined as vertex of [Formula: see text] represent an edge of [Formula: see text] and two vertices of [Formula: see text] are adjacent if and only if their respective edges share a common endpoint in [Formula: see text]. In this article, we examine graph theoretic properties of the line graph [Formula: see text] of [Formula: see text] and obtain some results related to completeness, diameter, weak perfectness and girth of this graph.