The rank of a graph after vertex addition

Bevis, Jean H.; Blount, Kevin K.; Davis, George J.; Domke, Gayla S.; Miller, Valerie A.

doi:10.1016/s0024-3795(96)00513-7

Cited by 33 publications

(16 citation statements)

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“…A simple and special case consists in taking a node p and joining a chain of length 2 to it, that is, connect p with a new node p 1 and that node in turn with another new node p 2 and put the value 0 at p 1 and the value −f 1 (p) at p 2 . This case was obtained in [1].…”

Section: Proof a Corresponding Eigenfunction Is Obtained Asmentioning

confidence: 86%

“…Local operations like adding an edge may increase or decrease m 1 or leave it invariant. Adding a pending vertex to a chain of length 2 increases m 1 from 0 to 1, adding a pending vertex to closed chain of length 3, a triangle, leaves m 1 = 0, adding a pending vertex to a closed chain of length 4, a quadrangle, reduces m 1 from 2 to 1 (see [1] for general results in this direction). Similarly, closing a chain by adding an edge between the first and last vertex may increase, decrease or leave m 1 the same.…”

Section: Examplesmentioning

confidence: 99%

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On the spectrum of the normalized graph Laplacian

Banerjee

Jost

2008

Linear Algebra and its Applications

110

139

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The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under local and global operations like motif doubling, graph joining or splitting. The eigenvalue 1 plays a particular role, and we therefore emphasize those constructions that change its multiplicity in a controlled manner, like the iterated duplication of nodes.

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Section: Proof a Corresponding Eigenfunction Is Obtained Asmentioning

confidence: 86%

Section: Examplesmentioning

confidence: 99%

On the spectrum of the normalized graph Laplacian

Banerjee

Jost

2008

Linear Algebra and its Applications

110

139

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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: 72%

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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“…When it is not essential to specify α explicitly, we write G ⊕ α v simply as G ⊕ v. Note that α can be viewed as the characteristic function of the neighborhood N (v) of v in G ⊕ α v with respect to vertex set V (G). The exact effect on the rank of a graph when a single vertex is added has been examined in [3]. 3.…”

Section: Elamentioning

confidence: 99%