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 , 2 k 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 x 1 ′ x 2 ′ of G , w t ( x 1 x 2 ) ≠ w t ( x 1 ′ x 2 ′ ) , where w t ( x 1 x 2 ) = f v ( x 1 ) + f e ( x 1 x 2 ) + f v ( x 2 ) . 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.