On the metric dimension, line graph and some graphs that can be considered as the zero divisor graph of a poset
Subject Areas : Algebra
1 - Department of Mathematics, University of Neyshabur, Neyshabur, Iran
Keywords: Partially ordered set, Zero divisor graph, Line graph, Graph with a horn, Metric dimension.,
Abstract :
Let (P,) be a partially ordered set with the least element 0. The zero divisor graph of P , denoted by (P) , is a simple and undirected graph whose vertex set is the set of all nonzero zero divisors of P and two distinct vertices x and y are adjacent if and only if {x , y }l {0} , where for a vertex z , the set {z }l is the lower bound of z . The line graph of , which is denoted by , is a graph whose vertex set is equal to the set of all edges of and two distinct vertices are adjacent if and only if their corresponding edges in have a vertex in common. is called a line graph if there exists a graph such that is isomorphic to . In this paper, we study the line zero divisor graph of a partially ordered set and we determine all partially ordered sets with at most three atoms such that their is a line graph. Also, we characterize all connected graphs with at most five vertices and all complete graphs and complete bipartite graphs with horns, that can be considered as the zero divisor graph of a partially ordered set. Moreover, we study the metric dimension of .
