A characterization of generalized cospectrality of rooted graphs with applications in graph reconstruction

TL;DR

This paper extends classical spectral graph theory results to characterize generalized cospectrality in rooted graphs, providing new insights into graph reconstructibility based on vertex-deleted subgraphs and automorphisms.

Contribution

It offers a matrix-based characterization of generalized cospectral graphs and introduces a novel reconstructibility condition involving almost controllable subgraphs with automorphisms.

Findings

Matrix characterization of generalized cospectrality

New reconstructibility condition for graphs with automorphisms

Reconstructibility proven for graphs with certain vertex-deleted subgraphs

Abstract

Extending a classic result of Johnson and Newman, this paper provides a matrix characterization for two generalized cospectral graphs with a pair of generalized cospectral vertex-deleted subgraphs. As an application, we present a new condition for the reconstructibility of a graph. In particular, we show that a graph with at least three vertices is reconstructible if there exists a vertex-deleted subgraph that is almost controllable and has a nontrivial automorphism.