On Distance Irregularity Strength of Lollipop, Centipede, and Tadpole Graphs

Abstract

Let G be a simple graph. A distance irregular vertex k-labelling of a graph G is defined as a labelling λ:V(G)⟶{1,2,…,k} which is every two distinct vertices x,y∈V(G) have different weights, wt(x)≠wt(y). The weight of a vertex x in G, denoted by wt(x), is the sum of the labels of all the vertices adjacent to x (distance 1 from x), namely, wt(x)= ∑y∈N(x)λ(y), where N(x) is the set of all the vertices adjacent to x. The minimum k for which the graph G has a distance irregular vertex k-labelling is called the distance irregularity strength of G and denoted by dis(G). In this paper, we determine the exact value of the distance irregularity strength of lollipop, tadpole, and centipede graphs