Equitable irregular edge-weighting of graphs

Abstract

Given a graph $G=(V,E)$, an $k$-edge-weighting is a map $\varphi :E(G)\to \{1,2,3,\dots ,k\}$, where $k$ is a positive integer. For a vertex $v$ of $G$, let $S_{\varphi }(v)$ denote the sum of edge-weights appearing on the edges incident at $v$ under the edge-weighting $\varphi$. An $k$-edge-weighting of $G$ is said to be equitable irregular if $|S_{\varphi }(u)-S_{\varphi }(v)|\le 1$, for every pair of adjacent vertices $u$ and $v$ in $G$. A graph $G$ is said to be equitable irregular if $G$ admits an equitable irregular edge-weighting. If $G$ is equitable irregular then the equitable irregular strength of $G$ is dened to be the smallest positive integer $k$ such that there is a $k$-edge-weighting of $G$, and is denoted by $Se(G)$. In this paper we initiate a study of this new non-proper edge weighting of graphs.