A note on line graphs

Abstract

The line graph and 1-quasitotal graph are well-known concepts in graph theory. In Satyanarayana, Srinivasulu, and Syam Prasad [13], it is proved that if a graph G consists of exactly m connected components Gi (1 ≤ i ≤ m) then L(G) = L(G1) = L(G2) ⊕ ... ⊕ L(Gm) where L(G) denotes the line graph of G, and ⊕ denotes the ring sum operation on graphs. In [13], the authors also introduced the concept 1- quasitotal graph and obtained that Q1(G) = G⊕L(G) where Q1(G) denotes 1-quasitotal graph of a given graph G. In this note, we consider zero divisor graph of a finite associate ring R and we will prove that the line graph of Kn−1 contains the complete graph on n vertices where n is the number of elements in the ring R.