On inclusive distance vertex irregular labelings
Martin Bača, Andrea Semaničová–Feňovčíková, Slamin Slamin, Kiki Ariyanti Sugeng · Electronic Journal of Graph Theory and Applications · 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 w t f ( 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 w t f ( u ) ≠ w t f ( 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.