27 research outputs found
A Maximum Principle for Combinatorial Yamabe Flow
This article studies a discrete geometric structure on triangulated manifolds
and an associated curvature flow (combinatorial Yamabe flow). The associated
evolution of curvature appears to be like a heat equation on graphs, but it can
be shown to not satisfy the maximum principle. The notion of a parabolic-like
operator is introduced as an operator which satisfies the maximum principle,
but may not be parabolic in the usual sense of operators on graphs. A maximum
principle is derived for the curvature of combinatorial Yamabe flow under
certain assumptions on the triangulation, and hence the heat operator is shown
to be parabolic-like. The maximum principle then allows a characterization of
the curvature as well was a proof of long term existence of the flow.Comment: 20 pages, this is an almost entirely different paper. Some elements
of the old version are in the paper arxiv:math.MG/050618
Green functions for boundary value problems on product networks
Postprint (published version
Forest matrices around the Laplacian matrix
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