Degree distance and minimum degree

Abstract

Let $G$ be a finite connected graph of order $n$, minimum degree $\delta$ and diameter $d$. The degree distance $D^\prime(G)$ of $G$ is defined as $\sum_{\{u,v\}\subseteq V(G)}({\rm deg}~ u+{\rm deg}~ v)d(u,v)$, where ${\rm deg}~ w$ is the degree of vertex $w$ and $d(u,v)$ denotes the distance between $u$ and $v$. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that $$D^\prime(G)\le \frac{1}{4}dn\left(n-\frac{d}{3}(\delta+1)\right)^2+O(n^3).$$ As a corollary, we obtain the bound $D^\prime(G)\le \frac{n^4}{9(\delta+1)}+O(n^3)$ for a graph $G$ of order $n$ and minimum degree $\delta$. This result, apart from improving on a result of Dankelmann, Gutman, Mukwembi and Swart [3] for graphs of given order and minimum degree, settles a conjecture of Tomescu [14] completely. Let $G$ be a finite connected graph of order $n$, minimum degree $\delta$ and diameter $d$. The degree distance $D^\prime(G)$ of $G$ is defined as $\sum_{\{u,v\}\subseteq V(G)}(\operatorname{deg} u+\operatorname{deg}v)d(u,v)$, where $\operatorname{deg}w$ is the degree of vertex $w$ and $d(u,v)$ denotes the distance between $u$ and $v$. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that $$D^\prime(G)\le \frac{1}{4}dn\left(n-\frac{d}{3}(\delta+1)\right)^2+O(n^3).$$ As a corollary, we obtain the bound $D^\prime(G)\le \frac{n^4}{9(\delta+1)}+O(n^3)$ for a graph $G$ of order $n$ and minimum degree $\delta$. This result, apart from improving on a result of Dankelmann, Gutman, Mukwembi and Swart [3] for graphs of given order and minimum degree, settles a conjecture of Tomescu [14] completely. doi:10.1017/S000497271200035