Edge irregular reflexive labeling of some tree graphs

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 f_e from the edge set to the first k_e natural number, and a function f_v from the vertex set to the non negative even number up to 2k_v, where k = max k_e, 2k_v }. 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 =f_v x₁ +f_e x₁ x₂ + f_v(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.