Non-Inclusive and Inclusive Distance Irregularity Strength of Complement and Split Graphs

Abstract

Let $f$ be a map from vertices of a graph $G$ to number from $1$ to $k$. The labeling $f$ is called distance irregular if for every two vertices $x$ and $y$, it holds that $wt_f(u) \ne wt_f(v)$ where a weight $wt_f(u)$ is defined as the sum of labels of the neighbors of $u$. Moreover, the labeling $f$ is called inclusive distance irregular if for every two vertices $x$ and $y$, $wt_f(u) \ne wt_f(v)$ with a weight $wt_f(u)$ is defined as the sum of the label of $u$ and the labels of the neighbors of $u$. The least number $k$ where there exists a distance irregular labeling (resp. inclusive distance irregular labeling) is called a distance irregularity strength (inclusive distance irregularity strength), denoted by $\text{dis}(G)$ $(\widehat{\text{dis}}(G))$. In this paper, we present a connection of distance irregular labeling and inclusive distance irregular labeling in a graph with its complement. In particular, we derive a new upper bound for distance irregularity strength and inclusive distance irregularity strength of any graph. Further, we determine the $\text{dis}(G)$ and $\widehat{\text{dis}}(G)$ for certain special family of split graph $G$ and provide examples of a graph $G$ satisfying $\text{dis}(G) = \widehat{\text{dis}}(G)$.