Generalized matrix tree theorem for mixed graphs

Bapat, R.B.; Grossman, Jerrold W.; Kulkarni, Amey

doi:10.1080/03081089908818623

Cited by 51 publications

(40 citation statements)

References 5 publications

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“…Set a ij = sgn (v i , v j ) if (v i , v j ) ∈ E(G) and a ij = 0 otherwise. Then A(G) = [a ij ] is called the adjacency matrix of G. The Laplacian matrix of G is defined as L(G) = D(G) + A(G) [1,19], where D(G) = diag {d(v 1 ), d(v 2 ), . .…”

Section: Introductionmentioning

confidence: 99%

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Fan

Gong

Wang

et al. 2009

Discrete Mathematics

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a b s t r a c tLet G be a mixed graph and let L(G) be the Laplacian matrix of the graph G. The first eigenvalue and the first eigenvectors of G are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of L(G). In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order.

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Section: Introductionmentioning

confidence: 99%

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Fan

Gong

Wang

et al. 2009

Discrete Mathematics

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“…After row condensation with respect to T 1 (add rows 2, 5 to row 4, then delete rows 2,5) and column condensation with respect to T 2 (add columns 3, 4 to column 1, then delete columns 3, 4) we get the following matrix.…”

Section: L(ij|uv)| = (−1)mentioning

confidence: 99%

On minors of the compound matrix of a Laplacian

Bapat

2013

Linear Algebra and its Applications

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“…The Laplacian matrix [48] is defined as Q = ( in + out ), in , and out are diagonal matrices, which contain the in-degree and out-degree of each node respectively, andĀ = 1 2 (A + A T ). If the network is an undirected network,Ā is the adjacency matrix A and = diag (d 1 ,d 2 , .…”

Section: Algebraic Connectivity Of Directed Networkmentioning

confidence: 99%

Epidemic threshold in directed networks

2013

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Epidemics have so far been mostly studied in undirected networks. However, many real-world networks, such as the online social network Twitter and the world wide web, on which information, emotion, or malware spreads, are directed networks, composed of both unidirectional links and bidirectional links. We define the directionality ξ as the percentage of unidirectional links. The epidemic threshold τ c for the susceptible-infected-susceptible (SIS) epidemic is lower bounded by 1/λ 1 in directed networks, where λ 1 , also called the spectral radius, is the largest eigenvalue of the adjacency matrix. In this work, we propose two algorithms to generate directed networks with a given directionality ξ . The effect of ξ on the spectral radius λ 1 , principal eigenvector x 1 , spectral gap (λ 1 − |λ 2 |), and algebraic connectivity μ N−1 is studied. Important findings are that the spectral radius λ 1 decreases with the directionality ξ , whereas the spectral gap and the algebraic connectivity increase with the directionality ξ . The extent of the decrease of the spectral radius depends on both the degree distribution and the degree-degree correlation ρ D . Hence, in directed networks, the epidemic threshold is larger and a random walk converges to its steady state faster than that in undirected networks with the same degree distribution.

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Generalized matrix tree theorem for mixed graphs

Cited by 51 publications

References 5 publications

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

On minors of the compound matrix of a Laplacian

Epidemic threshold in directed networks

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