Local distance irregular labeling of graphs

Abstract

We introduce the notion of distance irregular labeling, called the local distance irregular labeling. We define ? : V (G) ?? {1, 2, . . . , k} such that the weight calculated at the vertices induces a vertex coloring if w(u) 6? w(v) for any edge uv. The weight of a vertex u ? V (G) is defined as the sum of the labels of all vertices adjacent to u (distance 1 from u), that is w(u) = ?y?N(u)?(y). The minimum cardinality of the largest label over all such irregular assignment is called the local distance irregularity strength, denoted by disl(G). In this paper, we found the lower bound of the local distance irregularity strength of graphs G and also exact values of some classes of graphs namely path, cycle, star graph, complete graph, (n, m)-tadpole graph, unicycle with two pendant, binary tree graph, complete bipartite graphs, sun graph.