The reflexive edge strength on some almost regular graphs

Abstract

A function f with domain and range are respectively the edge set of graph G and natural number up to $k_e$ , and a function f with domain and range are respectively the vertex set of graph G and the even natural number up to $2k_v$ are called a total k-labeling where $k=max\{k_e,2k_v\}$ . The total k-labeling of graph G by the condition that every two different edges have different weight is called an edge irregular reflexive k-labeling, where for any edge $x_1{x}_2$ , the weight is $wt\left(x_1{x}_2\right)=f_v\left(x_1\right)+f_e\left(x_1{x}_2\right)+f_v\left(x_2\right)$ . The reflexive edge strength of the graph G, denoted by $res\left(G\right)$ is the minimum k for graph G which has an edge irregular reflexive k-labelling. In this study, we obtained the $res\left(G\right)$ of graphs which their vertex degrees show an almost regularity properties.

Keywords: P n × C 3 ; P n ⊙ C 3 ; P n ⊙ P 2 ; Edge irregular reflexive k-labeling; Ladder graph; Reflexive edge strength; Triangular ladder.