The Maximal Total Irregularity of Unicyclic Graphs

Abstract

In $[1]$, Hosam Abdo and Darko Dimitrov introduced the total irregularity of a graph. For a graph $G$, it is defined as
$${\text{irr}}_t(G)=\frac{1}{2}\sum _{u,v\in V(G)}|d_G(u)–d_G(v)|,$$
where $d_G(u)$ denotes the vertex degree of a vertex $u\in V(G)$. In this paper, we introduce two transformations to study the total irregularity of unicyclic graphs and determine the graph with the maximal total irregularity among all unicyclic graphs with $n$ vertices.