On inclusive distance vertex irregular labelings - Martin Baca; Andrea Semanicova-Fenovcikova; S. Slamin; Kiki A. Sugeng (Electronic Journal of Graph Theory and Applications (EJGTA), 2018)
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 wtf(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 wtf(u) ≠ wtf(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.