具有最小全不规则度的非正则图
Non-regular graphs with minimal total irregularity
摘要：
简单无向图$G$的{ \it全不规则度}定义为${\rm irr}_t(G) =$ $\frac{1}{2}\sum_{u,v \in V(G)}$ $\left| d_G(u)-d_G(v) \right|$，其中$d_G(u)$表示顶点$u \in V(G)$的度。显然，${\rm irr}_t(G)=0$当且仅当$G$是正则的。本文刻画了具有最小全不规则度的非正则连通图，从而解决了最近由Zhu，You和Yang~ \cite{zyy-mtig-2014}提出的关于非正则连通图的最小全不规则度下界的猜想.证明了$2n-4$的严格下界仅在偶数阶非正则连通图上才能得到，而$n-1$的严格下界则在奇数阶非正则连通图上才能得到.我们还刻画了具有第二和第三小总不规则性的非正则图。
英文摘要：
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.
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