On inclusive distance vertex irregularity strength of small identical copies of star graphs

Abstract

For a simple graph G, an inclusive distance vertex irregular k-labeling of G is a mapping λ : V (G) → {1,2,…, k} such that all the vertex-weights are pairwise distinct, where the weight of a vertex v, denoted by wt(v), is the sum of labels of vertices in the close neighborhood of the vertex v. The minimum k for which the graph G has an inclusive distance vertex irregular k-labeling is called the inclusive distance vertex irregularity strength of G, d i s ^ ( G ) . Here we introduce a new lower bound for d i s ^ ( G ) and determine the exact value of the inclusive distance vertex irregularity strength for identical copies of star graphs, especially 2Sn and 3Sn .

An Inclusive Distance Irregularity Strength of n-ary Tree

“…The exact value of the inclusive distance vertex irregularity strength of many graphs have been widely studied in the literature (see Bača et al, 2018;Utami et al, 2018;Bong et al, 2020;Halikin et al, 2020;Utami et al, 2020;Susanto et al, 2021;Majid et al, 2023;and Windartini et al, 2014). Furthermore, Cichacz et al (2021) gave the upper bound of the inclusive distance vertex irregularity strength for a simple graph G on n vertices in which no two vertices have the same closed neighborhood that dis ̂(𝐺) ≤ 𝑛 2 .…”

“…Santoso et al (2022) using genetic algorithm for the inclusive labeling of a graph. Bong et al (2020) gave a lower bound for the inclusive distance vertex irregularity strength for a graph G of order n, the maximum degree ∆, and the minimum degree δ, Susanto et al (2021) developed a new lower bound for the inclusive distance vertex irregularity strength of graphs that generalizes the lower bound by (Bong et al, 2020), that a graph G with the maximum degree ∆, the minimum degree δ,…”

Non-inclusive and inclusive distance irregularity strength for the join product of graphs

“…A number of research results on non-inclusive and inclusive d-distance irregularity strengths have been found as seen in [3,4,11,13,14,15,16,17] when d = 1 and in [5,18] when d > 1.…”