Related papers: Counting Spanning Trees on Fractal Graphs
The connection between the maximum spanning tree in a directed graph and the best dependency tree of a sentence has been exploited by the NLP community. However, for many dependency parsing schemes, an important detail of this approach is…
The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these…
In this paper, we study some spanning trees with bounded degree and leaf degree from eigenvalues. For any integer $k\geq2$, a $k$-tree is a spanning tree in which every vertex has degree no more than $k$. Let $T$ be a spanning tree of a…
We consider the number of spanning trees in circulant graphs of $\beta n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $\beta n$ factors, while we derive a formula of $\beta-1$…
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…
Exact eigendecomposition of large matrices is very expensive, and it is practically impossible to compute exact eigenvalues. Instead, one may set a more modest goal of approaching the empirical distribution of the eigenvalues, recovering…
In the present paper, we investigate a family of circulant graphs with non-fixed jumps $$G_n=C_{\beta n}(s_1, \ldots,s_k,\alpha_1n,\ldots,\alpha_\ell n),\, 1\le s_1<\ldots<s_k\le[\frac{\beta n}{2}],\, 1\le \alpha_1<…
In this paper, we investigate the problem of generating the spanning trees of a graph $G$ up to the automorphisms or "symmetries" of $G$. After introducing and surveying this problem for general input graphs, we present algorithms that…
A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite…
In this paper, we aim to design sparse D-optimal (determinantoptimal) pose-graph SLAM problems through the synthesis of sparse graphs with the maximum weighted number of spanning trees. Characterizing graphs with the maximum number of…
In this work, we study the Anderson model on the Sierpinski gasket graph. We first identify the almost sure spectrum of the Anderson model when the support of the random potential has no gaps. We then prove the existence of the integrated…
We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$, provided that the number of edges is at least $n + \textstyle{\frac{1}{2}}…
In this paper, we compute asymptotics for the determinant of the combinatorial Laplacian on a sequence of $d$-dimensional orthotope square lattices as the number of vertices in each dimension grows at the same rate. It is related to the…
A graph can be regarded as an electrical network in which each edge is a resistor. This point of view relates combinatorial quantities, such as the number of spanning trees, to electrical ones such as effective resistance. The second and…
We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization…
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…
We prove the existence of an upper bound on the asymptotic dimension of tree amalgamations of locally finite quasi-transitive connected graphs. This generalises a result of Dranishnikov for free products with amalgamation and a result of…
We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient…
We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree…
We consider the polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less…