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 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.

The reflexive edge strength on some almost regular graphs

“…In the following theorems, we show the results on edge irregular reflexive k -labeling and reflexive edge strength of some almost regular graphs, namely ladder, triangular ladder, , , and . To prove each theorem, we use lemma of the lower bound for which is appeared in [12] and [13] .…”

“…For every n , it gives . We refer to lemmas in [12] and [13] to determine the lower bound of . …”

“…Let , , be a graph with the vertex set , and the edge set . We refer to lemmas in [12] and [13] to determine the lower bound of . …”

“…The smallest value of k for which such labeling exists is called the reflexive edge strength of the graph G , denoted by [12] . Some Lemmas of the lower bound for can be seen in [12] , [13] , [14] .…”

“…Ibrahim et al in [15] determined the , where G were star, double star , and caterpillar graphs. Agustin et al in [13] determined the , where G were generalized subdivided star, broom, and double star graph . Bača et al in [14] determined the , where G was cycle, cartesian product of cycles, join of cycle graphs and .…”