The Spectrum of Triangle-free Graphs
J\'ozsef Balogh, Felix Christian Clemen, Bernard Lidick\'y, Sergey Norin, Jan Volec

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 the smallest eigenvalue of the signless Laplacian matrix of an -vertex graph . Brandt conjectured in 1997 that for regular triangle-free graphs . We prove a stronger result: If is a triangle-free graph then . Brandt's conjecture is a subproblem of two famous conjectures of Erd\H{o}s: (1) Sparse-Half-Conjecture: Every -vertex triangle-free graph has a subset of vertices of size spanning at most edges. (2) Every -vertex triangle-free graph can be made bipartite by removing at most edges. In our proof we use linear algebraic methods to upper bound 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research · Synthesis and Properties of Aromatic Compounds
