Related papers: Counting spanning trees in self-similar networks b…
The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform…
In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic…
In this paper, we introduce two families of planar and self-similar graphs which have small-world properties. The constructed models are based on an iterative process where each step of a certain formulation of modules results in a final…
In this paper algebraic and combinatorial properties and a computation of the number of the spanning trees are developed for certain graphs. To this purpose, an original method, independent of the spectrum of the Laplacian matrix associated…
A network-theoretic approach for determining the complexity of a graph is proposed. This approach is based on the relationship between the linear algebra (theory of determinants) and the graph theory. In this paper we contribute a new…
In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological…
The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial…
We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures…
Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such…
Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral…
Spanning trees are an important quantity characterizing the reliability of a network, however, explicitly determining the number of spanning trees in networks is a theoretical challenge. In this paper, we study the number of spanning trees…
The weighted spanning tree enumerator of a graph $G$ with weighted edges is the sum of the products of edge weights over all the spanning trees in $G$. In the special case that all of the edge weights equal $1$, the weighted spanning tree…
Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be…
We derive the determinant of the Laplacian for the Hanoi networks and use it to determine their number of spanning trees (or graph complexity) asymptotically. While spanning trees generally proliferate with increasing average degree, the…
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…
The complete knowledge of Laplacian eigenvalues and eigenvectors of complex networks plays an outstanding role in understanding various dynamical processes running on them; however, determining analytically Laplacian eigenvalues and…
Construction of phylogenetic trees and networks for extant species from their characters represents one of the key problems in phylogenomics. While solution to this problem is not always uniquely defined and there exist multiple methods for…
Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a…
The eigenvalues of the normalized Laplacian matrix of a network plays an important role in its structural and dynamical aspects associated with the network. In this paper, we study the spectra and their applications of normalized Laplacian…
(DRAFT VERSION) In this article we present a proof of the famous Kirchoff's Matrix-Tree theorem, which relates the number of spanning trees in a connected graph with the cofactors (and eigenvalues) of its combinatorial Laplacian matrix.…