Measuring Similarity Between Connected Graphs: The Role of Induced Subgraphs and Complementarity Eigenvalues

Abstract

This work elaborates on the old problem of measuring the degree of similarity, say f(G,H), between a pair of connected graphs G and H, not necessarily of the same order. The choice of a similarity index f depends essentially on the graph properties that are considered as important in a given context. As relevant information on a graph, one may consider for instance its degree sequence, its characteristic polynomial, and so on. We explore some new similarity indices based on nonstandard spectral information contained in the graphs under comparison. By nonstandard spectral information in a graph, we mean the set of complementarity eigenvalues of the adjacency matrix. From such a spectral perspective, two distinct graphs G and H are viewed as highly similar if they share a large number of complementarity eigenvalues. This basic idea will be cast in a rigorous mathematical formalism.