Some properties on the lexicographic product of graphs obtained by monogenic semigroups

Ismail Naci Cangul,Ahmet Sinan Cevik,Kinkar C Das,Nihat Akgunes

doi:10.1186/1029-242x-2013-238

Ismail Naci Cangul, Ahmet Sinan Cevik

Open Access

https://doi.org/10.1186/1029-242x-2013-238

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Abstract

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph on monogenic semigroups (with zero) having elements was recently defined. The vertices are the non-zero elements and, for , any two distinct vertices and are adjacent if in . As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randic index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over were investigated by the same authors of this paper. In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over . In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs and . MSC:05C10, 05C12, 06A07, 15A18, 15A36.

Highlights

Introduction and preliminariesThe base of the graph Γ (SM) is zero-divisor graphs
In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over Γ (SM)
We investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two graphs Γ (SM1 ) and Γ (SM2 )

Summary

Introduction

Introduction and preliminariesThe base of the graph Γ (SM) is zero-divisor graphs (cf. [ ]). We investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs Γ (SM1 ) and Γ (SM2 ). The lexicographic product G[H] of any two simple graphs G and H (in some references, it is called composition product [ ]) is defined which has the vertex set V (G) × V (H) such that any two vertices u = (u , u ) and v = (v , v ) are connected to each other by an edge if and only if u v ∈ E(G) or u = v and u v ∈ E(H) (see, for instance, [ – ]).

Results

Conclusion