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On inclusive distance vertex irregular labelings

Abstract:

For a simple graph G, a vertex labeling f: V (G) → (1; 2....,k) is called a k-labeling. The weight of a vertex v, denoted by wt<inf>f</inf> (v) is the sum of all vertex labels of vertices in the closed neighborhood of the vertex v. A vertex k-labeling is defined to be an inclusive distance vertex irregular distance k-labeling of G if for every two different vertices u and v there is wt<inf>f</inf> (u) ≠ wt<inf>f</inf> (v). The minimum k for which the graph G has a vertex irregular distance k-labeling is called the inclusive distance vertex irregularity strength of G. In this paper we establish a lower bound of the inclusive distance vertex irregularity strength for any graph and determine the exact value of this parameter for several families of graphs.