Inertia of complex unit gain graphs

Yu, Guihai; Qu, Hui; Tu, Jianhua

doi:10.1016/j.amc.2015.05.105

Cited by 21 publications

(18 citation statements)

References 14 publications

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“…It will be left to our future study. In addition, there are many spectrum-based invariants, which are widely investigated, such as graph energy (e.g., graph theory [19,20], incidence energy [21], and matching energy [22,23]), HOMO-LUMO index [24,25], and inertia [26][27][28][29]. In the future, we would like to study some properties of these spectrum-based indices of signed networks.…”

Section: Resultsmentioning

confidence: 99%

More on Spectral Analysis of Signed Networks

2018

Complexity

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Spectral graph theory plays a key role in analyzing the structure of social (signed) networks. In this paper we continue to study some properties of (normalized) Laplacian matrix of signed networks. Sufficient and necessary conditions for the singularity of Laplacian matrix are given. We determine the correspondence between the balance of signed network and the singularity of its Laplacian matrix. An expression of the determinant of Laplacian matrix is present. The symmetry about 1 of eigenvalues of normalized Laplacian matrix is discussed. We determine that the integer 2 is an eigenvalue of normalized Laplacian matrix if and only if the signed network is balanced and bipartite. Finally an expression of the coefficient of normalized Laplacian characteristic polynomial is present.

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Section: Resultsmentioning

confidence: 99%

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2018

Complexity

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“…Yu et al [13] proved the following result. The complete graph K n is a well-known DS graph (see [2], Proposition 1 and Proposition 6).…”

Section: Elementary Lemmasmentioning

confidence: 91%

“…Reff [10] defined the adjacency, incidence and Laplacian matrices of a complex unit gain graph and gave eigenvalue bounds for the adjacency and Laplacian matrices of such graphs. Yu et al [13] obtained properties of inertia indexes of T -gain graphs and characterised the T -gain unicyclic graphs with small positive or negative index. Lu et al [7] characterised the T -gain connected bicyclic graphs with rank 2, 3 or 4.…”

Section: Introductionmentioning

confidence: 99%

Graphs Determined by Their -Gain Spectra

Wang

Wong²,

Tian³

2020

Bull. Aust. Math. Soc.

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An undirected graph $G$ is determined by its $T$ -gain spectrum (DTS) if every $T$ -gain graph cospectral to $G$ is switching equivalent to $G$ . We show that the complete graph $K_{n}$ and the graph $K_{n}-e$ obtained by deleting an edge from $K_{n}$ are DTS, the star $K_{1,n}$ is DTS if and only if $n\leq 2$ , and an odd path $P_{2m+1}$ is not DTS if $m\geq 2$ . We give an operation for constructing cospectral $T$ -gain graphs and apply it to show that a tree of arbitrary order (at least $5$ ) is not DTS.

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“…Lu et al [11] studied the relation between the rank of a complex unit gain graph and the rank of its underlying graph. In [21], the positive inertia and negative inertia of a complex unit gain cycle were characterized by Yu et al In [19] and Wang et al investigated the determinant of the Laplacian matrix of a complex unit gain graph. In [16], Reff generalized some fundamental concepts from spectral graph theory to complex unit gain graphs and defined the adjacency, incidence and Laplacian matrices of them.…”

Section: Introductionmentioning

confidence: 99%

“…Reff extended some fundamental concepts from spectral graph theory to complex unit gain graphs and defined the adjacency, incidence and Laplacian matrices of them in [16]. Yu et al [20] investigated some properties of inertia of complex unit gain graphs and discussed the inertia index of a complex unit gain cycle. In [19], Wang et al provided a combinatorial description of the determinant of the Laplacian matrix of a complex unit gain graph which generalized that for the determinant of the Laplacian matrix of a signed graph.…”

Section: Introductionmentioning

confidence: 99%

The rank of a complex unit gain graph in terms of the matching number

Hao

Dong

2020

Linear Algebra and its Applications

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A complex unit gain graph (or T-gain graph) is a triple Φ = (G, T, ϕ) ((G, ϕ) for short) consisting of a graph G as the underlying graph of (G, ϕ), T = {z ∈ C : |z| = 1} is a subgroup of the multiplicative group of all nonzero complex numbers C × and a gain function ϕ :− → E → T such that ϕ(e ij ) = ϕ(e ji ) −1 = ϕ(e ji ). In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph (G, ϕ) with order n, and prove that 2n − 2c(G) ≤ r(G, ϕ) + 2α(G) ≤ 2n. Where r(G, ϕ), α(G) and c(G) are the rank of the Hermitian adjacency matrix A(G, ϕ), the independence number and the cyclomatic number of G, respectively. Furthermore, the properties of the complex unit gain graph that reaching the lower bound are characterized.

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Inertia of complex unit gain graphs

Cited by 21 publications

References 14 publications

More on Spectral Analysis of Signed Networks

More on Spectral Analysis of Signed Networks

Graphs Determined by Their -Gain Spectra

The rank of a complex unit gain graph in terms of the matching number

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