Non-regular graphs with minimal total irregularity

Abstract

The {\it total irregularity} of a simple undirected graph $G$ is defined as ${\mathrm{irr}}_t(G)=$ $\frac{1}{2}{\sum }_{u,v\in V(G)}$ $\left|d_G(u)-d_G(v)\right|$, where $d_G(u)$ denotes the degree of a vertex $u\in V(G)$. Obviously, ${\mathrm{irr}}_t(G)=0$ if and only if $G$ is regular. Here, we characterize the non-regular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu, You and Yang~\cite{zyy-mtig-2014} about the lower bound on the minimal total irregularity of non-regular connected graphs. We show that the conjectured lower bound of $2n-4$ is attained only if non-regular connected graphs of even order are considered, while the sharp lower bound of $n-1$ is attained by graphs of odd order. We also characterize the non-regular graphs with the second and the third smallest total irregularity.