On Valuation of Edge Irregularity Strength of Certain Graphical Families
Zhiqiang Zhang, Tariq Mehmood, Atiq ur Rehman, Muhammad Tahir Hussain, Xiujun Zhang · Journal of Mathematics · 2022
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(vi) + δ(vj), i ≠ j = 1,2, …, n, defines the weight wδ(vivj) of corresponding edge for every vivj ∈ E. If two different edges ei and ej in graph G carry weights in different manner, i.e., wδ(ei) ≠ wδ((ei) 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.