Edge irregular reflexive labeling of some tree graphs

Abstract

Let G be a connected, simple, and undirected graph with a vertex set V(G) and an edge set E(G). Total k-labeling is a function fe from the edge set to the first ke natural number, and a function fv from the vertex set to the non negative even number up to 2kv, where k = max ke, 2kv }. An edge irregular reflexive k labeling of the graph G is the total k-labeling, if for every two different edges x 1 x 2 and x1' x2' of G,wt(x1 x2 ≠ wt(x1' x2' where wt(x1 x2 =fv x1 +fe x1 x2 + fv(x2). The minimum k for graph G which has an edge irregular reflexive k-labelling is called the reflexive edge strength of the graph G, denoted by res(G). In this paper, we determined the exact value of the reflexive edge strength of family trees, namely generalized sub-divided star graph, broom graphs, and double star graph.