The spectral reconstruction problem revisited

Abstract

This paper focuses on the following spectral reconstruction problem: Can a graph be uniquely determined (up to isomorphism) by the collection of its spectrum and the spectra of its vertex-deleted graphs? We say two graphs are hyper-cospectral if they share identical spectrum and identical spectra for their vertex-deleted subgraphs. A graph G is spectrally reconstructible (SRC for short) if any graph H that is hyper-cospectral with G is also isomorphic to G. Tutte [13] showed that graphs with irreducible characteristic polynomials are SRC. We aim to extend this result to a larger family of graphs known as controllable graphs. Since not all controllable graphs are SRC, we address the problem: Which controllable graphs are SRC? We provide a proof for a family of controllable bipartite graphs whose characteristic polynomials have exactly two irreducible factors. Moreover, we demonstrate that for a non-controllable graph G with a characteristic polynomial having exactly two irreducible factors, if a hyper-cospectral graph exists, then their complements are also hyper-cospectral. In addition, we also present an algorithm for determining the spectral reconstructibility of graphs whose characteristic polynomials split into exactly two irreducible factors, as well as for finding their hyper-cospectral mates when they exist. Some examples are also provided to illustrate our results.