Refutation of Spectral Graph Theory Conjectures with Search Algorithms)

TL;DR

A wide range of search algorithms are applied to a selection of conjectures from Graffiti to find potentially large counter-examples to spectral graph theory conjectures in seconds.

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.

Figures (7)

• Figure 1: A counter-example of Graffiti 289 of size 20 (second largest eigenvalue $\le$ mean of the mean of all adjacent vertex degree for all nodes)
• Figure 2: A counter-example of Graffiti 301 of size 14 (scope of positive eigenvalues $\le$ harmonic)
• Figure 3: A counter-example of Graffiti 30 of size 15 (# positive distance eigenvalues $\le$ sum of temperatures)
• Figure 4: A counter-example of Graffiti 29 of size 7 (randic index $\le$ # negative eigenvalues)
• Figure 5: A counter-example of Graffiti 137 of size 67 (second largest eigenvalue $\le$ harmonic)