On degree powers and counting stars in $F$-free graphs

TL;DR

The paper tackles the problem of maximizing $e_r(G)=\sum_{i=1}^n d_i^r$ over $n$-vertex $F$-free graphs, linking degree-powered sums to generalized Turán problems. A key method expresses $e_r(G)$ as a weighted sum of star counts $\mathcal N(S_p,G)$, enabling a close connection to $\mathrm{ex}(n,S_r,F)$ and the decomposition family $\mathcal D(F)$ via $\mathrm{biex}(n,F)$. The authors prove exact extremal results for graphs with a color-critical edge, showing $\mathrm{ex}_r(n,F)=e_r(T)$ for a complete $(\chi(F)-1)$-partite graph $T$, and obtain several non-bipartite cases as well as $\mathrm{ex}_r(n, C_4)$ for $\,r\ge 3$. They also develop a stability framework that yields precise structural descriptions and extends known results with simpler star-counting proofs, bridging degree-based indices and generalized Turán theory with broad applicability, including chemical graph theory contexts.

Abstract

Given a positive integer $r$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_r(G)=\sum_{i=1}^n d_i^r$. We let $\mathrm{ex}_r(n,F)$ be the largest value of $e_r(G)$ if $G$ is an $n$-vertex $F$-free graph. We show that if $F$ has a color-critical edge, then $\mathrm{ex}_r(n,F)=e_r(G)$ for a complete $(χ(F)-1)$-partite graph $G$ (this was known for cliques and $C_5$). We obtain exact results for several other non-bipartite graphs and also determine $\mathrm{ex}_r(n,C_4)$ for $r\ge 3$. We also give simple proofs of multiple known results. Our key observation is the connection to $\mathrm{ex}(n,S_r,F)$, which is the largest number of copies of $S_r$ in $n$-vertex $F$-free graphs, where $S_r$ is the star with $r$ leaves. We explore this connection and apply methods from the study of $\mathrm{ex}(n,S_r,F)$ to prove our results. We also obtain several new results on $\mathrm{ex}(n,S_r,F)$.

Theorems & Definitions (31)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4: Nikiforov, nik
• Theorem 1.5: Chang, Chen and Zhang, ccz
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• proof : New proof of Theorem \ref{['niki']}