NONREGULAR GRAPHS WITH MINIMAL TOTAL IRREGULARITY

Abstract

Thetotal irregularityof 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 Zhuet 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.