Degree powers in graphs with a forbidden forest

TL;DR

This paper extends Turán-type extremal graph results to degree power sums for graphs excluding certain forests, providing exact bounds for large n.

Contribution

It generalizes previous results by determining degree power sum extremal functions for linear, star, and broom forests in large graphs.

Findings

Determined $ex_p(n,F)$ for linear forests $F$ for large $n$.

Calculated $ex_p(n,S)$ for star forests.

Established bounds for broom graphs with diameter at most six.

Abstract

Given a positive integer $p$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: Given a positive integer $p$ and a graph $H$, determine the function $ex_p(n,H)$, which is the maximum value of $e_p(G)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Tur\'an number. Caro and Yuster determined the function $ex_p(n, P_\ell)$ for sufficiently large $n$, where $p\geq 2$ and $P_\ell$ denotes the path on $\ell$ vertices. In this paper, we generalise this result and determine $ex_p(n,F)$ for sufficiently large $n$, where $p\geq 2$ and $F$ is a linear forest. We also determine $ex_p(n,S)$, where $S$ is a star forest; and $ex_p(n,B)$, where $B$ is a broom graph with diameter at most six.