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Distance irregularity strength of graphs with pendant vertices

Susanto Faisal, Wijaya Kristiana, Slamin, Semaničová-Feňovčíková Andrea

Opuscula Mathematica

2022

Vol. 42, no. 3

439--458

A vertex k-labeling ϕ : V (G) → {1, 2, . . . , k} on a simple graph G is said to be a distance irregular vertex k-labeling of G if the weights of all vertices of G are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in G. The least integer k for which G has a distance irregular vertex k-labeling is called the distance irregularity strength of G and denoted by dis(G). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper.