A NOTE ON LINE GRAPHS

• Published in: TWMS journal of applied and engineering mathematics Vol. 7; no. 1; p. 173
• Main Authors: Bhavanari, S, Devanaboina, S, Kuncham, S.P
• Format: Journal Article
• Language: English
• Published: Istanbul Turkic World Mathematical Society 01.06.2017; Elman Hasanoglu
• Subjects: Graph theory; Graphs; Mathematical research; Rings (Algebra)
• ISSN: 2146-1147; 2146-1147

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 [G.sub.i] (1 [less than or equal to] i [less than or equal to] m) then L(G) = L([G.sub.1]) = L([G.sub.2]) [symmetry]... [symmetry] L([G.sub.m]) where L(G) denotes the line graph of G, and [symmetry] denotes the ring sum operation on graphs. In [13], the authors also introduced the concept 1-quasitotal graph and obtained that [Q.sub.1](G) = G [symmetry] L(G) where [Q.sub.1](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 [K.sub.n-1] contains the complete graph on n vertices where n is the number of elements in the ring R. Keywords: line graph, quasi-total graph, zero-divisor graph, associate ring, complete graph. AMS Subject Classification: 05C25, 05C76, 05C99, 13E15, 68R10.