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 EIS , symbolized as eis G , 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 δ : V ⟶ 1,2 , … , p 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., δ v i + δ v j , i ≠ j = 1,2 , … , n , defines the weight w δ v i v j of corresponding edge for every v i v j ∈ E . If two different edges e i and e j in graph G carry weights in different manner, i.e., w δ e i ≠ w δ e i for i ≠ 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.