Localised graph Maclaurin inequalities

TL;DR

This paper extends the Maclaurin inequalities for graphs by incorporating a weight function to analyze local structures, settling a recent conjecture and generalizing previous localized results.

Contribution

It introduces a weighted extension of the graph Maclaurin inequalities that captures local graph structure and proves a conjecture by Kirsch and Nir.

Findings

Extended Maclaurin inequalities with local weights

Settled a recent conjecture of Kirsch and Nir

Unified previous localized results

Abstract

The Maclaurin inequalities for graphs are a broad generalisation of the classical theorems of Tur\'an and Zykov. In a nutshell they provide an asymptotically sharp answer to the following question: what is the maximum number of cliques of size $q$ in a $K_{r+1}$-free graph with a given number of cliques of size $s$? We prove an extensions of the graph Maclaurin inequalities with a weight function that captures the local structure of the graph. As a corollary, we settle a recent conjecture of Kirsch and Nir, which simultaneously encompass the previous localised results of Brada\v{c}, Malec and Tompkins and of Kirsch and Nir.