Characterizing cospectral vertices via isospectral reduction

Abstract

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices. *

Designing pretty good state transfer via isospectral reductions

“…For this reason, graphs (or, just as well, matrices) that lack direct symmetries, but whose eigenstates fulfill Eqs. (7) and (8) were recently termed latently symmetric [18,19]. However, although these symmetries may indeed seem hidden, we would like to mention here that all the graphs shown in Fig.…”

“…We thereby create the iconic graph that is shown in the first paper on cospectral vertices by Schwenk [33] and is also depicted in many publications related to cospectrality, for example in Refs. [11,19,32].…”

“…We first provide some key aspects of isospectral reductions [13][14][15][16][17]19], introduced first by Bunimovich and Webb [13]. This concept will allow us to extract the polynomials P ± .…”

“…Namely, by directly obtaining the polynomials P ± from an underlying symmetric Hamiltonian that features cospectral sites u and v. To this end, we combine the mathematical relations underlying the works in Ref. [11,12] with the theory of isospectral reductions [13][14][15][16][17][18][19]. Isospectral reduction is a method to reduce the size of a given Hamiltonian while keeping a large amount of information on its eigenvalues and eigenvectors.…”

“…We utilize the isospectral reduction to "compress" only the relevant spectral information for the problem at hand by building upon the very recent results of Ref. [19]. These results put strong constraints on the structure of the isospectral reduction of a Hamiltonian that features cospectral sites.…”