Hermitian adjacency matrix of the second kind for mixed graphs

Li, Shuchao; Yu, Yuantian

doi:10.1016/j.disc.2022.112798

Cited by 13 publications

(8 citation statements)

References 27 publications

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“…Inspired by the conclusions introduced by Mohar [1] and Shuchao Li et al [2] above, we find that ϕ(e ij ) ∈ T 3 in (1) is also feasible. Furthermore, we can study all the cases of ϕ(e ij ) ∈ T k (k ≥ 3).…”

Section: Introductionsupporting

confidence: 55%

“…By setting k = 6 and ω = cos( 2π 6 ) + i sin( 2π 6 ) in theorem 2.5, 2.1, we can get the two-way switching and three-way switching results in [2].…”

Section: V1mentioning

confidence: 99%

“…Mohar [1] introduced a new Hermitian matrix (N-matrix for short) with ϕ(e ij ) ∈ T 6 in (1) and studied the eigenvalues of this new matrix. Shuchao Li et al [2] investigate some basic properties of the N-matrix, which may be viewed as a continuance of Mohar's work [1]. They give some equivalent switchings, which are called two-way and three-way switchings, to get cospectral graphs for a mixed graph.…”

Section: Introductionmentioning

confidence: 99%

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On Hermitian Adjacency Matrices for Mixed Graphs

She¹,

Wang²

2022

Preprint

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We study the spectra of mixed graphs about its Hermitian adjacency matrix of the second kind (i.e. N-matrix) introduced by Mohar [1]. We extend some results and define one new Hermitian adjacency matrix, and the entry corresponding to an arc from u to v is equal to the k-th( or the third) root of unity, i.e. ω = cos( 2π k ) + i sin( 2π k ), k ≥ 3; the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. In this paper, we characterize the cospectrality conditions for mixed graphs with the same underlying graph. In section 4, we determine a sharp upper bound on the spectral radius of mixed graphs, and provide the corresponding extremal graphs.

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Section: Introductionsupporting

confidence: 55%

“…By setting k = 6 and ω = cos( 2π 6 ) + i sin( 2π 6 ) in theorem 2.5, 2.1, we can get the two-way switching and three-way switching results in [2].…”

Section: V1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Hermitian Adjacency Matrices for Mixed Graphs

She¹,

Wang²

2022

Preprint

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“…This is introduced by Mohar [28]. We refer to [2,13,25] as recent studies on Hermitian adjacency matrices. Note that H η (G ± ) coincides with the ordinary adjacency matrix of G ± .…”

Section: Preliminaries On Spectral Graph Theorymentioning

confidence: 99%

Periodicity of quantum walks defined by mixed paths and mixed cycles

Kubota,

Sekido,

Yata

2021

Preprint

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In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of η-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of η-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define n + 1 types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss periodicity. We show that the mixed paths are periodic for any η.In addition, we provide a necessary and sufficient condition for a mixed cycle to be periodic and determine their periods.

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“…This is introduced by Mohar [30]. We refer to [1,9,10,11,26,45] as recent studies on Hermitian adjacency matrices. Note that H η (G ± ) coincides with the ordinary adjacency matrix of G ± .…”

Section: Preliminariesmentioning

confidence: 99%

Combinatorial necessary conditions for regular graphs to induce periodic quantum walks

Kubota¹

2021

Preprint

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We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. If the quantum walk is periodic, all the eigenvalues of the time evolution matrices must be algebraic integers. Focusing on this, we explore which ring the coefficients of the characteristic polynomials should belong to. On the other hand, the coefficients of the characteristic polynomials of η-Hermitian adjacency matrices have combinatorial implications. From these, we can find combinatorial implications in the coefficients of the characteristic polynomials of the time evolution matrices, and thus derive combinatorial necessary conditions for mixed graphs to be periodic. For example, if a k-regular mixed graph with n vertices is periodic, then 2n/k must be an integer. As an application of this work, we determine periodicity of mixed complete graphs and mixed graphs with a prime number of vertices.

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Hermitian adjacency matrix of the second kind for mixed graphs

Cited by 13 publications

References 27 publications

On Hermitian Adjacency Matrices for Mixed Graphs

On Hermitian Adjacency Matrices for Mixed Graphs

Periodicity of quantum walks defined by mixed paths and mixed cycles

Combinatorial necessary conditions for regular graphs to induce periodic quantum walks

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