The Minimal Total Irregularity of Graphs

Abstract

In \cite{2012a}, Abdo and Dimitov defined the total irregularity of a graph $G=(V,E)$ as

\hskip3.3cm $\rm irr_{t}$$(G) = \frac{1}{2}\sum_{u,v\in V}|d_{G}(u)-d_{G}(v)|,  $

\noindent where $d_{G}(u)$
denotes the vertex degree of a vertex $u\in V$.
In this paper, we investigate the minimal total irregularity of the connected graphs,
determine the minimal, the second minimal, the third minimal total irregularity  of trees, unicyclic graphs, bicyclic graphs on $n$ vertices,
and propose an open problem for further research.