On inclusive 1-Distance Vertex Irregularity Strength of Firecracker, Broom, and Banana Tree

Abstract

Let k be a natural number and G be a simple graph. An inclusive d-distance vertex irregular labelling of a graph G is a function 𝜆: 𝑉(𝐺) ⟶ {1,2, … , 𝑘} so that the weights at each vertex are different. Let v be a vertex of G. The weight of v ∈V(G), denoted by wt(v), is the sum of the label of v and all vertex labels up to distance 1 from v. An inclusive 1-distance vertex irregularity strength of G, denoted by 𝑑𝑖𝑠 ̂ (𝐺) is the minimum k for the existence of an inclusive 1-distance vertex irregular labelling of a G. Here, we find the exact value of an inclusive 1-distance vertex irregularity strength of a firecracker, a broom, anda banana tree.