The skew-rank of oriented graphs

Li, Xueliang; Yu, Guihai

doi:10.1360/012014-51

Cited by 34 publications

(19 citation statements)

References 13 publications

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“…Then let [T G ] be obtained from T G by deleting all the contracted vertices and the incident edges (see [21] for details). Clearly, Lemmas 2.5, 2.6 and 2.7 can be easily deduced, respectively, the corresponding results for undirected graphs (see [11,5,3] for details).…”

Section: Some Known Lemmasmentioning

confidence: 71%

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Relation between the H-rank of a mixed graph and the rank of its underlying graph

Chen

Zhang

2019

Discrete Mathematics

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Given a simple graph G = Denote by P n , C n and K n a path, a cycle and a complete graph of order n, respectively. Let d(G) denote the dimension of cycle spaces of a graph G. Then d(G) = |E G | − |V G | + ω(G), here ω(G) is the number of connected components of G. Two distinct edges in a graph G are independent if they do not have a common end-vertex in G. A set of pairwise independent edges of G is called a matching, while a matching with the maximum cardinality is called a maximum matching. This maximum cardinality is called the matching number of G and written by m(G). A graph is called an empty graph if it has no edges.The adjacency matrix A(G) = (a ij ) of G is an n × n matrix whose a ij = 1 if vertices i and j are adjacent and 0 otherwise. The skew-adjacency matrix associated to an oriented graph G σ , written as S(G σ ), is defined to be an n × n matrix (s uv ) such that s uv = 1 if there is an arc from u to v, s uv = −1 if there is an arc from v to u and s uv = 0 otherwise. The Hermitian adjacency matrix of a mixed graph G is defined to be an n × n matrixotherwise. This matrix was introduced, independently, by Liu and Li [12] and Guo and Mohar [7]. Since H( G)is Hermitian, its eigenvalues are real. The H-rank (resp. rank, skew-rank) of G (resp. G, G σ ), denoted by rk( G) (resp. r(G), sr(G σ )), is the rank of H( G) (resp. A(G), S(G σ )).Recently, the study on the H-rank and the characteristic polynomial of mixed graphs attracts more and more researchers' attention. Mohar [15] characterized all the mixed graphs with H-rank 2 and showed that there are infinitely many mixed graphs with H-rank 2 which can not be determined by their H-spectrum. Wang et al. [20] identified all the mixed graphs with H-rank 3 and showed that all mixed graphs with H-rank 3 can be determined by their H-spectrum. Liu and Li [12] investigated the properties for characteristic polynomials of mixed graphs and studied the cospectral problems among mixed graphs. For more properties and applications about the H-rank and eigenvalues of mixed graphs, we refer the readers to [1,7,8,16] and the references therein.Note that, for a mixed graph G, it is possible that E G = E 0 G or E G = E 1 G . Hence, both oriented graphs and simple graphs can be seen as the special mixed graphs. Wong, Ma and Tian [21] provided a beautiful relation between the skew-rank of an oriented graph and the rank of its underlying graph, which were extended by Huang and Li [9]. Recently, Ma, Wong and Tian [14] determined the relationship between sr(G σ ) and the matching number m(G), whereas in [13] they characterized the relationship between rank of G and its number of pendant vertices, from which it may deduce the relationship between the skew rank of G σ and its number of pendant vertices. Huang, Li and Wang [10] established the relationship between sr(G σ ) and the independence number of its underlying graph G. Very recently, Chen, Huang and Li [4] studied the relation between the H-rank of a

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Section: Some Known Lemmasmentioning

confidence: 71%

“…In order to characterize the extremal graphs whose H-rank attains upper and lower bounds in (1.1), we first introduce a graph transformation (see also in [11] and [21]).…”

mentioning

confidence: 99%

Relation between the H-rank of a mixed graph and the rank of its underlying graph

Chen

Zhang

2019

Discrete Mathematics

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“…For G 16 , let the rows of S(G σ 16 ) correspond successively the vertices v 3 , v 2 , v 4 , v 6 , v 5 , v 1 , v 9 , v 7 , v 8 , v 10 , v 11 , v 12 . It follows that,…”

Section: All Connected 5-regular Optimum Skew Energy Oriented Graphsmentioning

confidence: 99%

“…(it is G 17 ), G 20 has an optimum orientation. Next we give the skew adjacency matrix of G σ 17 with rows corresponding to v 1 , v 10…”

Section: All Connected 5-regular Optimum Skew Energy Oriented Graphsmentioning

confidence: 99%

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5-regular oriented graphs with optimum skew energy

Guo

Wang

Xiao

2017

Applied Mathematics and Computation

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Let G be a simple undirected graph and G σ be the corresponding oriented graph of G with the orientation σ. The skew energy of G σ , denoted by ε s (G σ ), is defined as the sum of the singular values of the skew adjacency matrix S(G σ ). In 2010, Adigawhere ∆ is the maximum degree of G of order n. In this paper, we determine all connected 5-regular oriented graphs of order n with maximum skew-energy.

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