The Minimal Total Irregularity of some Classes of Graphs

Yingxue Zhu, Lihua You, Jieshan Yang

Abdo and Dimitov defined the total irregularity of a graph $G=(V,E)$ as $\operatorname{irr}_t(G)=\frac12\sum_{u,v\in V}|d_G(u)-d_G(v)|$, 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.