Computing total edge irregularity strength for heptagonal snake graph and related graphs

F Salama

doi:10.1007/s00500-021-06364-2

Abstract

A labeling of edges and vertices of a simple graph \(G(V,E)\) by a mapping \(\Lambda :V\left( G \right) \cup E\left( G \right) \to \left\{ { 1,2,3, \ldots ,\Psi } \right\}\) provided that any two pair of edges have distinct weights is called an edge irregular total \(\Psi\)-labeling. If \(\Psi\) is minimum and \(G\) admits an edge irregular total \(\Psi\) -labelling, then \(\Psi\) is called the total edge irregularity strength (TEIS) and denoted by \(\mathrm{tes}\left(G\right).\) In this paper, we start by defining new families of graphs called heptagonal snake graph \( {\mathrm{HPS}}_{\mathrm{n}} \),the double heptagonal snake graph \( D({\mathrm{HPS}}_{\mathrm{n}}), \) and an \(l-\) multiple heptagonal snake graph\( L({\mathrm{HPS}}_{\mathrm{n}}) \). We follow some steps to deduce the exact value of TEISs for the new families. We first labeled the vertices, and then the edges were labeled such that the weights of edges are different. After that, we calculated that exact value of TEISs for the new families.

Highlights

Graph labeling is an assignment of labels or weights to the vertices and/or edges of a graph
Graph labeling plays an important role in many fields such as computer science, coding theory, astronomy and physics
Main results: we introduce the definition of the heptagonal snake graph HPSn and we deduce the exact value of total edge irregularity strength (TEIS) for it

Summary

Introduction

Graph labeling is an assignment of labels or weights to the vertices and/or edges of a graph. Ћ} has been defined in [1] as a labeling of its vertices and edges such that for any pair of edges pq and p∗q∗ in a graph G their weights are distinct, i.e.wŦ(pq) ≠ wŦ(p∗q∗) where wŦ(pq) = Ŧ(pq) + Ŧ(p) + Ŧ(q). The authors in [1] introduced an inequality that gives bounds of TEIS of a graph G tes(G) ≥ max {⌈|E(G3)|+2⌉ , ⌈∆G2+1⌉}. In [2], Ivanĉo and Jendroî has deduced TEIS for a tree They introduced the following conjecture Conjecture 1. The definitions of the heptagonal snake graph HPSn ,the double heptagonal snake graph D(HPSn) and an l −multiple heptagonal snake graph L(HPSn) have been introduced. The exact value of TEISs for a heptagonal snake graph, a double heptagonal snake graph and an l −multiple heptagonal snake graph has been investigated

Computing TEIS of heptagonal snake graph HPSn

Conclusion