The Spectrum of Triangle-free Graphs

TL;DR

This paper proves a new upper bound on the smallest eigenvalue of the signless Laplacian for triangle-free graphs, strengthening a conjecture and connecting to Erdős's famous conjectures using algebraic and flag algebra methods.

Contribution

It establishes a tighter bound on the eigenvalue for triangle-free graphs and employs novel algebraic techniques and flag algebra methods to do so.

Findings

Proved that for any triangle-free graph, q_n(G) rac{15n}{94}

Connected the eigenvalue bound to Erdf6s's conjectures on triangle-free graphs

Used linear algebra and flag algebra techniques to derive bounds

Abstract

Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \leq \frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) \leq \frac{15n}{94}< \frac{4n}{25}$. Brandt's conjecture is a subproblem of two famous conjectures of Erd\H{o}s: (1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $\lceil\frac{n}{2}\rceil$ spanning at most $n^2/50$ edges. (2) Every $n$-vertex triangle-free graph can be made bipartite by removing at most $n^2/25$ edges. In our proof we use linear algebraic methods to upper bound $q_n(G)$ by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.