On NP-hard graph properties characterized by the spectrum

TL;DR

This paper investigates the computational complexity of graph properties characterized by spectral properties, showing some are NP-hard to test and exploring spectral characterizations of NP-hard problems, including constructing cospectral graphs with different Hamiltonian properties.

Contribution

It demonstrates that spectral graph properties can be NP-hard to test and provides new spectral characterizations of certain NP-hard problems, including explicit constructions.

Findings

Testing some spectral properties is NP-hard.

Constructed pairs of cospectral graphs with different Hamiltonian properties.

Spectral characterizations can represent NP-hard problems.

Abstract

Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which testing whether a graph has that property can be NP-hard (or belong to other computational complexity classes consisting of even harder problems). In addition, we discuss a possible spectral characterization of some well-known NP-hard problems. In particular, for every integer $k\geq 6$ we construct a pair of $k$-regular cospectral graphs, where one graph is Hamiltonian and the other one not.