On $W$-bases of directed graphs

Fiedler, Miroslav; Sedláček, Jiří

doi:10.21136/cpm.1958.108301

Cited by 20 publications

(10 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By virtue of ( 14) and Remark 1, m L (1) = m Lc (0) − 1. The statements regarding L c follow from those about L; after that, (ii) and the assertions regarding P result from (10). The proof of (iv) is straightforward.…”

Section: ⊓ ⊔mentioning

confidence: 75%

“…The assertions about L c follow from (10). The geometric multiplicities of the eigenvalues are the same, since the transformation ( 11) is nondegenerate: (12) defines its inverse.…”

Section: Laplacian Matrices and Stochastic Matricesmentioning

confidence: 98%

“…According to the famous matrix-tree theorem for digraphs [21], the (i, j) cofactor of such a matrix L(Γ) is equal to the total weight of spanning trees converging to i in Γ, where the weight of a subgraph is the product of the weights of its arcs. Generalizations of this theorem have been obtained in [10] and [5,3]. We call the matrices that satisfy (1) and (2) Laplacian matrices.…”

Section: Introductionmentioning

confidence: 94%

See 2 more Smart Citations

On the spectra of nonsymmetric Laplacian matrices

Agaev

Chebotarev

2005

Linear Algebra and its Applications

166

121

View full text Add to dashboard Cite

A Laplacian matrix is a square real matrix with nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with the absolute values of the off-diagonal entries not exceeding 1/n, where n is the order of the matrix. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of a standardized Laplacian matrix are equal to the in-forest dimension of the corresponding digraph and one less than the in-forest dimension of the complementary digraph, respectively. These eigenvalues are semisimple. The spectrum of a standardized Laplacian matrix belongs to the meet of two closed disks, one centered at 1/n, another at 1-1/n, each having radius 1-1/n, and two closed angles, one bounded with two half-lines drawn from 1, another with two half-lines drawn from 0 through certain points. The imaginary parts of the eigenvalues are bounded from above by 1/(2n) cot(pi/2n); this maximum converges to 1/pi as n goes to infinity. Keywords: Laplacian matrix; Laplacian spectrum of graph; Weighted directed graph; Forest dimension of digraph; Stochastic matrixComment: 11 page

show abstract

Section: ⊓ ⊔mentioning

confidence: 75%

“…The assertions about L c follow from (10). The geometric multiplicities of the eigenvalues are the same, since the transformation ( 11) is nondegenerate: (12) defines its inverse.…”

Section: Laplacian Matrices and Stochastic Matricesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

On the spectra of nonsymmetric Laplacian matrices

Agaev

Chebotarev

2005

Linear Algebra and its Applications

166

121

View full text Add to dashboard Cite

show abstract

“…Fiedler and Sedláček [25] proved that the principal minor of the Laplacian matrix resulting by the removal of the rows and columns indexed by a set J is equal to the total weight of in-forests with |J | trees rooted at the vertices of J .…”

Section: Introductionmentioning

confidence: 99%

Forest matrices around the Laplacian matrix

Chebotarev

Agaev

2002

Linear Algebra and its Applications

View full text Add to dashboard Cite

We study the matrices Q_k of in-forests of a weighted digraph G and their connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of L. The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the Laplacian matrix, which provides an immediate proof of the Markov chain tree theorem. A source of these results is the fact that matrices Q_k are the matrix coefficients in the polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's matrices for -L. Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection; Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic Graph Theor

show abstract

“…Corollary 3.1 (Fiedler and Sedláček [12]) The number c(K r,s ) of spanning trees of the complete bipartite graph K r,s is…”

Section: Bipartite Trees and Tripartite Treesmentioning

confidence: 99%

A recursive algorithm for trees and forests

Guo

2017

Discrete Mathematics

View full text Add to dashboard Cite

Abstract. Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with k roots and that with k + 1 roots on {1, 2, . . . , n}. Classical formulas for counting various trees such as rooted trees, bipartite trees, tripartite trees, plane trees, k-ary plane trees, k-edge colored trees follow immediately from our recursive relations.

show abstract

On $W$-bases of directed graphs

Cited by 20 publications

References 1 publication

On the spectra of nonsymmetric Laplacian matrices

On the spectra of nonsymmetric Laplacian matrices

Forest matrices around the Laplacian matrix

A recursive algorithm for trees and forests

Contact Info

Product

Resources

About