On Valuation of Edge Irregularity Strength of Certain Graphical Families

Abstract

This article comprises of exact valuation of a graph parameter, known as the edge irregularity strength $\left(\text{EIS}\right)$ , symbolized as $\text{eis}\left(G\right)$ , of various graphical families such as middle graph of path graph, middle graph of cycle graph, snake graph (string 2), paramedian ladder, and complete $m$ -partite graphs. If $\delta :V⟶\left\{\mathrm{1,2},\dots ,p\right\}$ is a function defined on vertices of a graph that helps to determine different weights for every pair of edges, the least value of $p$ is the target. Thus, addition operation for allocated to vertices of an edge, i.e., $\delta \left(v_i\right)+\delta \left(v_j\right)$ , $i\ne j=\mathrm{1,2},\dots ,n$ , defines the weight $w_{\delta }\left(v_i{v}_j\right)$ of corresponding edge for every $v_i{v}_j\in E$ . If two different edges $e_i$ and $e_j$ in graph $G$ carry weights in different manner, i.e., $w_{\delta }\left(e_i\right)\ne w_{\delta }\left(\left(e_i\right)\right)$ for $i\ne j$ . Then the edge irregular $p$ -labeling is defined after a vertex $p$ -labeling of $G$ . After establishing various novel results and making some conclusions, an open problem is mentioned in the end.