Tropicalization of graph profiles

Author:

Blekherman Grigoriy,Raymond Annie,Singh Mohit,Thomas Rekha

Abstract

A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph profiles. We introduce the tropicalization of graph and hypergraph profiles. Tropicalization is a well-studied operation in algebraic geometry, which replaces a variety (the set of real or complex solutions to a finite set of algebraic equations) with its “combinatorial shadow”. We prove that the tropicalization of a graph profile is a closed convex cone, which still captures interesting combinatorial information. We explicitly compute these tropicalizations for arbitrary sets of complete and star hypergraphs. We show they are rational polyhedral cones even though the corresponding profiles are not even known to be semialgebraic in some of these cases. We then use tropicalization to prove strong restrictions on the power of the sums of squares method, equivalently Cauchy-Schwarz calculus, to test (which is weaker than certification) the validity of graph density inequalities. In particular, we show that sums of squares cannot test simple binomial graph density inequalities, or even their approximations. Small concrete examples of such inequalities are presented, and include the famous Blakley-Roy inequalities for paths of odd length. As a consequence, these simple inequalities cannot be written as a rational sum of squares of graph densities.

Funder

National Science Foundation

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference35 articles.

1. Graphs with maximal number of adjacent pairs of edges;Ahlswede, R.;Acta Math. Acad. Sci. Hungar.,1978

2. Logarithmic limit sets of real semi-algebraic sets;Alessandrini, Daniele;Adv. Geom.,2013

3. Tropical spectrahedra;Allamigeon, Xavier;Discrete Comput. Geom.,2020

4. Threshold graphs maximize homomorphism densities;Blekherman, Grigoriy,2020

5. Simple graph density inequalities with no sum of squares proofs;Blekherman, Grigoriy;Combinatorica,2020

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Power Mean Inequalities and Sums of Squares;Discrete & Computational Geometry;2024-06-23

2. Tropicalizing the Graph Profile of Some Almost-Stars;SIAM Journal on Discrete Mathematics;2024-04-22

3. A path forward: Tropicalization in extremal combinatorics;Advances in Mathematics;2022-10

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3