On Non Inclusive Distance Vertex Irregularity Strength of Tadpole and Path Corona Path Graphs

Abstract

Let 𝐺 = (𝑉, 𝐸) be a connected and simple graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺). A non inclusive distance vertex irregular labeling of a graph 𝐺 is a mapping of 𝜆 ∶ (𝑉, 𝐺) → {1, 2, … , 𝑘} such that the weights calculated for all vertices are distinct. The weight of a vertex 𝑣, under labeling 𝜆, denoted by 𝑤𝑡(𝑣), is defined as the sum of the label of all vertices adjacent to 𝑣 (distance 1 from 𝑣). A non inclusive distance vertex irregularity strength of graph 𝐺, denoted by 𝑑𝑖𝑠(𝐺), is the minimum value of the largest label 𝑘 over all such non inclusive distance vertex irregular labeling. In this research, we determined 𝑑𝑖𝑠(𝐺) from 𝑇𝑚,𝑛 graph with 𝑚 ≥ 3, 𝑚 odd, 𝑎𝑛𝑑 𝑛 ≥ 1 and 𝑃𝑛 ⊙ 𝑃𝑛 graph 𝑤𝑖𝑡ℎ 𝑛 ≥ 2 and 𝑛 even.