Generalized spectral characterizations of almost controllable graphs

TL;DR

This paper introduces a new class of graphs called almost controllable graphs and provides spectral characterizations for them, extending previous results from controllable graphs to a broader class.

Contribution

It defines almost controllable graphs and proves spectral characterization results, including conditions for these graphs to be uniquely determined by their spectra.

Findings

For almost controllable graphs, exactly two rational orthogonal matrices relate generalized cospectral graphs.

A simple criterion is established for an almost controllable graph to be determined by its generalized spectrum.

The results extend spectral characterization techniques from controllable to almost controllable graphs.

Abstract

Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined by its spectrum. In Wang~[J. Combin. Theory, Ser. B, 122 (2017):438-451], the author gave a simple arithmetic condition for a family of graphs being determined by their generalized spectra. However, the method applies only to a family of the so called \emph{controllable graphs}; it fails when the graphs are non-controllable. In this paper, we introduce a class of non-controllable graphs, called \emph{almost controllable graphs}, and prove that, for any pair of almost controllable graphs $G$ and $H$ that are generalized cospectral, there exist exactly two rational orthogonal matrices $Q$ with constant row sums such that $Q^{\rm T}A(G)Q=A(H)$, where $A(G)$ and $A(H)$ are the adjacency matrices of $G$ and $H$, respectively. The main ingredient of the proof is a use of the Binet-Cauchy formula. As an application, we obtain a simple criterion for an almost controllable graph $G$ to be determined by its generalized spectrum, which in some sense extends the corresponding result for controllable graphs.