Godsil-McKay switching for mixed and gain graphs over the circle group

Belardo, Francesco; Brunetti, Maurizio; Cavaleri, Matteo; Donno, Alfredo

doi:10.1016/j.laa.2020.04.025

Cited by 13 publications

(12 citation statements)

References 9 publications

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“…Many particular cases have been considered in literature: first of all, the spectral theory of complex unit gain graphs [23], that are gain graphs whose gain group is T = {z ∈ C : |z| = 1}. 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].…”

Section: Introductionmentioning

confidence: 99%

“…The main goal of the present paper is the introduction of routines, inspired by the Godsil-McKay switching [17] and its generalizations [2,3], in order to obtain pairs of G-cospectral gain graphs and/or pairs of π-cospectral gain graphs for some unitary representation π of the group G, as well as an analysis of the relationships between the properties that make a graph suitable for the one or the other switching.…”

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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Godsil-McKay switchings for gain graphs

Cavaleri¹,

Donno²,

Spessato³

2022

Preprint

Self Cite

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“…This allowed a development of a spectral theory of T-gain graphs [27]. In particular Acharya's characterization of balance [24] and Godsil-McKay switching [5] have been extended to T-gain graphs.…”

Section: Introductionmentioning

confidence: 99%

“…5 − 48x 3 ) • (x 18 − 24x 16 − 4x 15 + 224x 14 + 64x 13 − 1024x 12 − 368x 11 + 2352x 10 + 896x 9 − 2432x 8 − 768x 7 + 768x 6 ).On the other hand, the characteristic polynomial of π St⊗A (A (Γ,ψ ′ ) ) is:(x 9 − 12x 7 + 44x 5 − 48x 3 ) • (x 18 − 24x 16 + 4x 15 + 224x 14 − 64x 13 − 1024x 12 + 368x 11 + 2352x 10 − 896x 9 − 2432x 8 + 768x 7 + 768x 6 ).Therefore (Γ, ψ) and (Γ, ψ ′ ) are π St -cospectral but they are not π St⊗A -cospectral. Notice that the idea behind the construction is that the element 1 S 4 +(12)(34)−(12)−(34) ∈ CS 4 is in the kernel of the extension of π St to CS 4 (it is also in the kernel of the extension of the natural permutation representation, that is equivalent to π 0 ⊕ π St ).…”

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confidence: 99%

On cospectrality of gain graphs

Cavaleri,

Donno

2021

Preprint

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We define G-cospectrality of two G-gain graphs (Γ, ψ) and (Γ ′ , ψ ′ ), proving that it is a switching isomorphism invariant. When G is a finite group, we prove that G-cospectrality is equivalent to cospectrality with respect to all unitary representations of 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 can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph Γ with n vertices and m edges, is equal to the number of simultaneous conjugacy classes of the group G m−n+1 .We provide examples of G-cospectral non-switching isomorphic graphs and we prove that any gain graph on a cycle is determined by its G-spectrum. Moreover, we show that when 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.

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“…There exist several operations on signed and gain graphs leaving the spectrum unchanged (see [6,7]). As we already observed, the switching is one of them [37].…”

Section: Introductionmentioning

confidence: 99%

Characterizations of line graphs in signed and gain graphs

Cavaleri,

D'Angeli,

Donno

2021

Preprint

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We generalize three classical characterizations of line graphs to line graphs of signed and gain graphs: the Krausz's characterization, the Rooij and Wilf's characterization and the Beineke's characterization. In particular, we present a list of forbidden gain subgraphs characterizing the class of gain-line graphs. In the case of a signed graph whose underlying graph is a line graph, this list consists of exactly four signed graphs. Under the same hypothesis, we prove that a signed graph is the line graph of a signed graph if and only if its eigenvalues are either greater than −2, or less than 2, depending on which particular definition of line graph is adopted.

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Godsil-McKay switching for mixed and gain graphs over the circle group

Cited by 13 publications

References 9 publications

Godsil-McKay switchings for gain graphs

Godsil-McKay switchings for gain graphs

On cospectrality of gain graphs

Characterizations of line graphs in signed and gain graphs

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