Affiliation:
1. Università degli Studi Niccolò Cusano - Via Don Carlo Gnocchi , 3 00166 Roma , Italia
Abstract
Abstract
We define
G
G
-cospectrality of two
G
G
-gain graphs
(
Γ
,
ψ
)
\left(\Gamma ,\psi )
and
(
Γ
′
,
ψ
′
)
\left(\Gamma ^{\prime} ,\psi ^{\prime} )
, proving that it is a switching isomorphism invariant. When
G
G
is a finite group, we prove that
G
G
-cospectrality is equivalent to cospectrality with respect to all unitary representations of
G
G
. Moreover, we show that two connected gain graphs are switching equivalent if and only if the gains of their closed walks centered at an arbitrary vertex
v
v
can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph
Γ
\Gamma
with
n
n
vertices and
m
m
edges, is equal to the number of simultaneous conjugacy classes of the group
G
m
−
n
+
1
{G}^{m-n+1}
. We provide examples of
G
G
-cospectral switching nonisomorphic graphs and we prove that any gain graph on a cycle is determined by its
G
G
-spectrum. Moreover, we show that when
G
G
is a finite cyclic group, the cospectrality with respect to a faithful irreducible representation implies the cospectrality with respect to any other faithful irreducible representation, and that the same assertion is false in general.
Subject
Geometry and Topology,Algebra and Number Theory
Reference38 articles.
1. D. M. Cvetković, M. Doob, and H. Sachs, Spectra of graphs, Theory and Application, Third edition, Johann Ambrosius Barth, Heidelberg, 1995, ii+447 pp.
2. E. R. van Dam and W. H. Haemers, Which graphs are determined by their spectrum? Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002), Linear Algebra Appl. 373 (2003), 241–272.
3. H. H. Günthard and H. Primas, Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit Systemen konjugierter Bindungen, Helv. Chim. Acta 39 (1956), 1645–1653.
4. L. Collatz and U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg 21 (1957), 63–77.
5. A. J. Schwenk, Almost all trees are cospectral, in: New Directions in the Theory of Graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), Academic Press, New York, 1973, pp. 275–307.
Cited by
2 articles.
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