Two irregularity measures possessing high discriminatory ability

Abstract

An $n$-vertex graph whose degree set consists of exactly $n-1$ elements is called antiregular graph. Such type of graphs are usually considered opposite to the regular graphs. An irregularity measure ($IM$) of a connected graph $G$ is a non-negative graph invariant satisfying the property: $IM(G) = 0$ if and only if $G$ is regular. The total irregularity of a graph $G$, denoted by $irr_t(G)$, is defined as $irr_t(G)= \sum_{\{u,v\} \subseteq V(G)} |d_u - d_v|$ where $V(G)$ is the vertex set of $G$ and $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively. Antiregular graphs are the most nonregular graphs according to the irregularity measure $irr_t$; however, various non-antiregular graphs are also the most nonregular graphs with respect to this irregularity measure. In this note, two new irregularity measures having high discriminatory ability are devised. Only antiregular graphs are the most nonregular graphs according to the proposed measures.