Laplace Spectra of Orgraphs and Their Applications

Abstract

The Laplace matrix is the matrix L = ( ij ) ∈ R n×n with nonpositive off-diagonal elements and zero row sums. A weighted orgraph corresponds to each Laplace matrix, its properties being closely related to the algebraic properties of the Laplace matrix. The normalized Laplace matrix L is the Laplace matrix where − 1 n ij 0 for all i = j. The paper was devoted to the spectrum of the Laplace matrices and to the relationship between the spectra of the Laplace and stochastic matrices. The normalized Laplace matrices were proved to be semiconvergent. It was established that the multiplicity of the eigenvalue 0 of the matrix L is equal to the in-forest dimension of the corresponding orgraph, and the multiplicity of the eigenvalue 1 is one less than the in-forest dimension of the complementary orgraph. The spectra of the matrices L belong to the intersection of two circles of radius 1 − 1/n centered at the points 1/n and 1 − 1/n, respectively. Additionally, the domain that comprises them is included in the intersection of two angles (defined in the paper) with vertices 0 and 1 and the band |Im (z)| 1 2n cot π 2n (at the limit |Im(z)| < 1 π ). A polygon with all points being the eigenvalues of the normalized n-order Laplace matrices was constructed.

On Coxeter Spectral Study of Posets and a Digraph Isomorphism Problem

“…On the other hand, it is shown how the ideas from algebraic combinatorics and the representation theory of posets and algebras (see [2], [15], [21]) successfully apply to the Coxeter spectral classification of posets, digraphs, edge-bipartite graphs, and signed graphs, see [1], [3], [5]- [7], [19]- [22] for applications.…”

“…Following the spectral graph theory problems, a graph coloring techniques, and algebraic methods in graph and poset theory studied in [1], [3], [4], [11], [15], [16], we continue a Coxeter spectral study of finite posets and edge-bipartite graphs [5], [7], [16]- [22] (or signed graphs in the sense of Harary [9] and Zaslavsky [23]). We study a connection between properties of the Coxeter spectrum specc J ⊆ C and the digraph isomorphism problem for Hasse digraphs of positive and non-negative posets J, compare with [1] and [3].…”

“…We study a connection between properties of the Coxeter spectrum specc J ⊆ C and the digraph isomorphism problem for Hasse digraphs of positive and non-negative posets J, compare with [1] and [3].…”

“…Here we follow mainly the terminology introduced in [1] and [3]. Note that the digraphs D 1 and D 2 are not isomorphic and they are cospectral, because…”

A Nernst heat theorem for nonequilibrium jump processes

“…We collect here mathematical ingredients, essential for the derivation of the nonequilibrium Nernst heat theorem. More details make the subject of a separate paper [32], in particular for the derivation of the graphical representations starting with the matrix-forest theorem [33][34][35]. For an introduction to graphical methods for nonequilibrium purposes, see [36].…”