On the sum of powers of the degrees of graphs

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. © 2012 Australian Mathematical Publishing Association Inc.