Non-regular graphs with minimal total irregularity

TL;DR

This paper characterizes non-regular graphs with minimal total irregularity, confirming a conjecture and identifying exact lower bounds for graphs of even and odd order.

Contribution

It resolves a recent conjecture by establishing the minimal total irregularity for non-regular graphs and characterizes those with second and third smallest irregularity.

Findings

Minimal total irregularity for even order graphs is 2n-4.

Minimal total irregularity for odd order graphs is n-1.

Confirmed the conjectured lower bound for non-regular connected graphs.

Abstract

The {\it total irregularity} of a simple undirected graph $G$ is defined as ${\rm irr}_t(G) =$ $\frac{1}{2}\sum_{u,v \in V(G)}$ $\left| d_G(u)-d_G(v) \right|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. Obviously, ${\rm irr}_t(G)=0$ 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.