Forest matrices around the Laplacian matrix
نویسندگان
چکیده
منابع مشابه
Forest matrices around the Laplacian matrix
We study the matrices Qk of in-forests of a weighted digraph Γ and their connections with the Laplacian matrix L of Γ. The (i, j) entry of Qk 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, Qk, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representation...
متن کاملComputing the additive degree-Kirchhoff index with the Laplacian matrix
For any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.
متن کاملLaplacian and the Adjacency Matrices
Proof. We first recall that every non-singular matrix B can be written B = QR, where Q is an orthonormal matrix Q and R is upper-triangular matrix R with positive diagonals1 We will use a slight variation of this fact, writing B = RQ. Now, since QT = Q−1, QAQT has exactly the same eigenvalues as A. Let Rt be the matrix t ∗R+ (1− t)I, and consider the family of matrices Mt = RtQAQR t , as t goes...
متن کاملSandpile groups and Laplacian matrices
Proof: (R ⊂ κ(G, s)). Let I ⊂ M(G, s) be a nonempty ideal. Fix ψ ∈ I. Given σ ∈ R, there exists a sandpile φ such that σ = (φ + ψ)◦ by defintion. Since I is an ideal, this implies that σ ∈ I. Hence R ⊂ I. We conclude that R ⊂ ⋂ I ideal of M(G,s) I = κ(G, s). (κ(G, s) ⊂ R). Recall that R 6= ∅. Since κ(G, s) is the minimal ideal of M(G, s), it is enough to show that R is an ideal. Consider σ ∈ R ...
متن کاملRandom matrices and Laplacian growth
The theory of random matrices with eigenvalues distributed in the complex plane and more general “β-ensembles” (logarithmic gases in 2D) is reviewed. The distribution and correlations of the eigenvalues are investigated in the large N limit. It is shown that in this limit the model is mathematically equivalent to a class of diffusion-controlled growth models for viscous flows in the Hele-Shaw c...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2002
ISSN: 0024-3795
DOI: 10.1016/s0024-3795(02)00388-9