ON THE SUM OF POWERS OF THE DEGREES OF GRAPHS
Renyu Xu, Jianliang Wu, Guanghui Wang, Xin Zhang · Bulletin of the Australian Mathematical Society · 2013
Abstract For positive integers $p$ and $q$ , let ${ \mathcal{G} }_{p, q} $ be a class of graphs such that $\vert E(G)\vert \leq p\vert V(G)\vert - q$ for every $G\in { \mathcal{G} }_{p, q} $ . In this paper, we consider the sum of the $k\mathrm{th} $ powers of the degrees of the vertices of a graph $G\in { \mathcal{G} }_{p, q} $ with $\Delta (G)\geq 2p$ . We obtain an upper bound for this sum that is linear in ${\Delta }^{k- 1} $ . These graphs include the planar, 1-planar, $t$ -degenerate, outerplanar, and series-parallel graphs.