On the Wiener Index of the Dot Product Graph over Monogenic Semigroups

BÃ¼ÅŸra AydÄ±n,Ä°smail Naci CangÃ¼l,Nihat AkgÃ¼neÅŸ

doi:10.29020/nybg.ejpam.v13i5.3745

Abstract

Algebraic study of graphs is a relatively recent subject which arose in two main streams: One is named as the spectral graph theory and the second one deals with graphs over several algebraic structures. Topological graph indices are widely-used tools in especially molecular graph theory and mathematical chemistry due to their time and money saving applications. The Wiener index is one of these indices which is equal to the sum of distances between all pairs of vertices in a connected graph. The graph over the nite dot product of monogenic semigroups has recently been dened and in this paper, some results on the Wiener index of the dot product graph over monogenic semigroups are given.

Highlights

Introduction and preliminariesThe connections between graph theory and ring theory was first established in 1988 by Beck [1]
Akgunes has defined the graph of monogenic semigroups in [11] in a similar manner
The Wiener index is one of the significant and surely the oldest topological indices used in mathematical chemistry

Summary

Introduction and preliminaries

The connections between graph theory and ring theory was first established in 1988 by Beck [1]. Badawi has studied the dot product graphs of commutative rings in [10]. The dot product graphs of monogenic semigroups have been studied in [12]. The Wiener index is one of the significant and surely the oldest topological indices used in mathematical chemistry. Let us recall the definitions of the Wiener index and monogenic semigroups: The vertex set of a graph G is denoted by V (G). The dot product graph (S) can be defined as an (undirected) graph with vertices X, Y ∈ S∗ = S \ {0S} such that X and Y are adjacent iff X · Y = 0Sn and this is denoted by X ∼ Y.

Main results

Examples