NONREGULAR GRAPHS WITH MINIMAL TOTAL IRREGULARITY

Abstract

The total irregularity of a simple undirected graph $G$ 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 degree of a vertex $u\in V(G)$. Obviously, $\text{irr}_{t}(G)=0$ if and only if $G$ is regular. Here, we characterise the nonregular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu et al. [‘The minimal total irregularity of graphs’, Preprint, 2014, arXiv:1404.0931v1 ] about the lower bound on the minimal total irregularity of nonregular connected graphs. We show that the conjectured lower bound of $2n-4$ is attained only if nonregular connected graphs of even order are considered, while the sharp lower bound of $n-1$ is attained by graphs of odd order. We also characterise the nonregular graphs with the second and the third smallest total irregularity.