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.