On the relation between the<i>H</i>-rank of a mixed graph and the matching number of its underlying graph

Chen, Chen; Huang, Jing; Li, Shuchao

doi:10.1080/03081087.2017.1374327

Cited by 41 publications

(21 citation statements)

References 20 publications

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“…We note that while the exact nature of the edges in Figure 6 may differ depending on u, the reasoning below remains valid. 6 Suppose that D is reduced. By Let us now consider digraphs that do contain T − a , T − b , or K − .…”

Section: Classification Of Digraphs With One Negative Eigenvaluementioning

confidence: 99%

The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum

Wissing

Dam

2020

Journal of Combinatorial Theory, Series A

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Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it is isomorphic to each digraph to which it is cospectral. Convincing numerical evidence to support the claim that this property is extremely rare is provided. Nonetheless, the first infinite family of connected digraphs that is SHDS is constructed. This family is obtained via the introduction of twin vertices into a structure that is named negative tetrahedron. This special digraph, that exhibits extreme spectral behavior, is contained in the surprisingly small collection of all digraphs with exactly one negative eigenvalue, which is determined as an intermediate result.

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Section: Classification Of Digraphs With One Negative Eigenvaluementioning

confidence: 99%

The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum

Wissing

Dam

2020

Journal of Combinatorial Theory, Series A

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“…Noting that all mixed forests with the same underlying graph have the same characteristic polynomial which is the characteristic polynomial of their underlying graph, we have To prove the necessity, it is enough to show that for any graph G not satisfying the conditions (1), (2) and 3, there exist two mixed graphs D and D in D(G) which are not switching equivalent, but cospectral. In fact, the Claims 1-4 imply the necessity.…”

Section: Odhs Graphsmentioning

confidence: 99%

“…to be the characteristic polynomial of D (D , resp.). Obviously, it is sufficient to prove that for any mixed graphs D and D with the same underlying graph G satisfying one of the conditions (1), (2) and 3 Thus, we may assume that s(D ) or s(D ) contains at least one real component. We divide the remaining proof into three cases according to the conditions of the theorem.…”

Section: Odhs Graphsmentioning

confidence: 99%

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On Graphs whose Orientations are Determined by their Hermitian Spectra

Wang¹,

Yuan²

2020

Electron. J. Combin.

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A mixed graph $D$ is obtained from a simple graph $G$, the underlying graph of $D$, by orienting some edges of $G$. A simple graph $G$ is said to be ODHS (all orientations of $G$ are determined by their $H$-spectra) if every two $H$-cospectral graphs in $\mathcal{D}(G)$ are switching equivalent to each other, where $\mathcal{D}(G)$ is the set of all mixed graphs with $G$ as their underlying graph. In this paper, we characterize all bicyclic ODHS graphs and construct infinitely many ODHS graphs whose cycle spaces are of dimension $k$. For a connected graph $G$ whose cycle space is of dimension $k$, we also obtain an achievable upper bound $2^{2k-1} + 2^{k-1}$ for the number of switching equivalence classes in $\mathcal{D}(G)$, which naturally is an upper bound of the number of cospectral classes in $\mathcal{D}(G)$. To achieve these, we propose a valid method to estimate the number of switching equivalence classes in $\mathcal{D}(G)$ based on the strong cycle basis, a special cycle basis introduced in this paper.

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“…In [4], the rank of the signed unicyclic graph was discussed by Fan et al He et al characterized the relation among the rank, the matching number and the cyclomatic number of a signed graph in [7]. Chen et al [3] investigated the relation between the H-rank of a mixed graph and the matching number of its underlying graph. For other research of the rank of a graph one may be referred to those in [1,9,14,15,18].…”

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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On the relation between theH-rank of a mixed graph and the matching number of its underlying graph

Cited by 41 publications

References 20 publications

The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum

The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum

On Graphs whose Orientations are Determined by their Hermitian Spectra

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

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