Refutation of Spectral Graph Theory Conjectures with Search Algorithms)

TL;DR

This paper introduces search algorithms to efficiently refute spectral graph theory conjectures, successfully refuting most known conjectures quickly and solving an open problem, outperforming previous exhaustive or reinforcement learning methods.

Contribution

It demonstrates the effectiveness of search algorithms in refuting spectral graph theory conjectures rapidly, including an open conjecture, surpassing prior exhaustive and reinforcement learning approaches.

Findings

Refuted 12 out of 13 conjectures from Graffiti in seconds.

Refuted an open conjecture (197) from Graffiti.

Search algorithms outperform previous methods in speed and capability.

Abstract

We are interested in the automatic refutation of spectral graph theory conjectures. Most existing works address this problem either with the exhaustive generation of graphs with a limited size or with deep reinforcement learning. Exhaustive generation is limited by the size of the generated graphs and deep reinforcement learning takes hours or days to refute a conjecture. We propose to use search algorithms to address these shortcomings to find potentially large counter-examples to spectral graph theory conjectures in seconds. We apply a wide range of search algorithms to a selection of conjectures from Graffiti. Out of 13 already refuted conjectures from Graffiti, our algorithms are able to refute 12 in seconds. We also refute conjecture 197 from Graffiti which was open until now.