The Minimal Total Irregularity of Graphs
  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. 
Yingxue Zhu, Lihua You, Jieshan Yang
math.CO