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Generalized spectral characterizations of almost controllable graphs | | Wei Wang
; Fenjin Liu
; Wei Wang
; | | Date: |
29 Oct 2020 | | 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
$Q^{
m T}A(G)Q=A(H)$, 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. | | Source: | arXiv, 2010.15888 | | Services: | Forum | Review | PDF | Favorites | |
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