Spectra of quaternion unit gain graphs

Belardo, Francesco; Brunetti, Maurizio; Coble, Nolan J.; Reff, Nathan; Skogman, Howard

doi:10.1016/j.laa.2021.09.009

Cited by 9 publications

(5 citation statements)

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“…Indeed, if q ′ = h −1 qh and Ax = xq, then the column vector xh satisfies the equation A(xh) = (xh)q ′ . Spectral theory of quaternionic matrices is a topic studied in the literature [5,15,20,30], also in relation with gain graphs [4]. It has even been shown that the analogue of the spectral theorem holds.…”

Section: Godsil-mckay Switching For Quaternion Unit Gain Graphsmentioning

confidence: 99%

“…Consider the U(H)-gain graphs (Γ, ψ) and (Γ, ψ) α,H of Examples 5.6 and 5.8. An explicit computation shows that the characteristic polynomials of the matrices f (A (Γ,ψ) ) and f (A (Γ,ψ) α,H ), or equivalently, the characteristic polynomials of the matrices π H (A (Γ,ψ) ) and π H (A (Γ,ψ) α,H ), coincide and are equal to x 16 − 22x 14 + 187x 12 − 776x 10 + 1639x 8 − 1650x 6 + 625x 4 and so the U(H)-gain graphs (Γ, ψ) and (Γ, ψ) α,H are right cospectral.…”

Section: Godsil-mckay Switching For Quaternion Unit Gain Graphsmentioning

confidence: 99%

“…Clearly, also in this case, complex matrices naturally arise and a spectral characterization of balance was proved [21], as well as an analogue of the Godsil-McKay switching [3]. Also the spectral theory for quaternion unit gain graphs was recently investigated [4], thanks to the existence of several notions of spectrum for quaternion valued matrices [5,15,20,30]. In [4] many of the classical results for signed and complex unit gain graphs are generalized to quaternion unit gain graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Also the spectral theory for quaternion unit gain graphs was recently investigated [4], thanks to the existence of several notions of spectrum for quaternion valued matrices [5,15,20,30]. In [4] many of the classical results for signed and complex unit gain graphs are generalized to quaternion unit gain graphs. One of the byproducts of the current paper is a generalization of the Godsil-McKay switching to this class of gain graphs.…”

Section: Introductionmentioning

confidence: 99%

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Godsil-McKay switchings for gain graphs

Cavaleri¹,

Donno²,

Spessato³

2022

Preprint

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We introduce a switching operation, inspired by the Godsil-McKay switching, in order to obtain pairs of G-cospectral gain graphs, that are gain graphs cospectral with respect to every representation of the gain group G. For instance, for two signed graphs, this notion of cospectrality is equivalent to the cospectrality of their signed adjacency matrices together with the cospectrality of their underlying graphs. Moreover, we introduce another more flexible switching in order to obtain pairs of gain graphs cospectral with respect to some fixed unitary representation. Many existing notions of spectrum for graphs and gain graphs are indeed special cases of these spectra associated with particular representations, therefore our construction recovers the classical Godsil-McKay switching and the Godsil-McKay switching for signed and complex unit gain graphs. As in the classical case, not all gain graphs are suitable for these switchings: we analyze the relationships between the properties that make the graph suitable for the one or the other switching. Finally we apply our construction in order to define a Godsil-McKay switching for the right spectrum of quaternion unit gain graphs.

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Section: Godsil-mckay Switching For Quaternion Unit Gain Graphsmentioning

confidence: 99%

Section: Godsil-mckay Switching For Quaternion Unit Gain Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Godsil-McKay switchings for gain graphs

Cavaleri¹,

Donno²,

Spessato³

2022

Preprint

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“…Belardo et al [1] studied the spectra of quaternion unit gain graphs. They defined the adjacency, Laplacian, and incidence matrices for a quaternion unit gain graph and study their properties.…”

mentioning

confidence: 99%

Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph

Zhou

2023

ELA

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Let $\Phi=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\Phi$, $U(\mathbb{Q})=\{z\in \mathbb{Q}: |z|=1\}$ is the circle group, and $\varphi:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. Let $A(\Phi)$ be the adjacency matrix of $\Phi$ and $r(\Phi)$ be the row left rank of $\Phi$. In this paper, we prove that $-2c(G)\leq r(\Phi)-r(G)\leq 2c(G)$, where $r(G)$ and $c(G)$ are the rank and the dimension of cycle space of $G$, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).

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