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Published
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Some relations between the skew spectrum of an oriented graph and the spectrum
of certain closely associated signed graphs
Volume 63, no. 1
(2022),
pp. 41–50
https://doi.org/10.33044/revuma.1914
Abstract
Let $R_{G'}$ be the vertex-edge incidence matrix of an
oriented graph $G'$. Let $\Lambda(\dot{F})$ be the signed graph whose
vertices are identified as the edges of a signed graph $\dot{F}$, with a pair of
vertices being adjacent by a positive (resp. negative) edge if and only if the
corresponding edges of $\dot{G}$ are adjacent and have the same
(resp. different) sign. In this paper, we prove that $G'$ is bipartite if and
only if there exists a signed graph $\dot{F}$ such that $R_{G'}^\intercal
R_{G'}-2I$ is the adjacency matrix of $\Lambda(\dot{F})$. It occurs that
$\dot{F}$ is fully determined by $G'$. As an application, in some particular
cases we express the skew eigenvalues of $G'$ in terms of the eigenvalues of
$\dot{F}$. We also establish some upper bounds for the skew spectral radius of
$G'$ in both the bipartite and the non-bipartite case.
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Published by the Unión Matemática Argentina |
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