On spectral properties for graph matching and graph isomorphism problems

TL;DR

This paper investigates the spectral conditions under which convex relaxations of graph matching problems are equivalent to the original problem, providing theoretical insights into graph isomorphism and automorphisms.

Contribution

It proves the equivalence of the graph matching problem and its convex relaxation for certain graphs based on spectral properties, advancing understanding of graph isomorphism.

Findings

Spectral properties determine when convex relaxations are exact

Characterization of automorphism groups via spectral analysis

Fundamental spectral properties of adjacency matrices derived

Abstract

Problems related to graph matching and isomorphisms are very important both from a theoretical and practical perspective, with applications ranging from image and video analysis to biological and biomedical problems. The graph matching problem is challenging from a computational point of view, and therefore different relaxations are commonly used. Although common relaxations techniques tend to work well for matching perfectly isomorphic graphs, it is not yet fully understood under which conditions the relaxed problem is guaranteed to obtain the correct answer. In this paper we prove that the graph matching problem and its most common convex relaxation, where the matching domain of permutation matrices is substituted with its convex hull of doubly-stochastic matrices, are equivalent for a certain class of graphs, such equivalence being based on spectral properties of the corresponding adjacency matrices. We also derive results about the automorphism group of a graph, and provide fundamental spectral properties of the adjacency matrix.