Eigenvalue location in graphs of small clique-width

TL;DR

This paper presents a method for efficiently locating eigenvalues of graphs with small clique-width by diagonalizing their adjacency matrices in polynomial time, given a clique-width expression.

Contribution

It introduces an algorithm for eigenvalue location in graphs of small clique-width that operates efficiently when a clique-width expression is provided.

Findings

Eigenvalues can be located in polynomial time for graphs with small clique-width.

Diagonalization of adjacency matrices is feasible in linear time relative to the graph size.

The method depends on having a known clique-width expression for the graph.

Abstract

Finding a diagonal matrix congruent to $A - cI$ for constants $c$, where $A$ is the adjacency matrix of a graph $G$ allows us to quickly tell the number of eigenvalues in a given interval. If $G$ has clique-width $k$ and a corresponding $k$-expression is known, then diagonalization can be done in time $O(\text{poly}(k) n)$ where $n$ is the order of $G$.