相关论文: Chip-Firing Games, $G$-Parking Functions, and an E…

We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm…

Kirchhoff's matrix tree theorem is a well-known result that gives a formula for the number of spanning trees in a finite, connected graph in terms of the graph Laplacian matrix. A closely related result is Wilson's algorithm for putting the…

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…

(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.…

Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e.,…

For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j…

The Laplacian matrix of a graph $G$ is $L(G)=D(G)-A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees. According to the Matrix-Tree Theorem, the number of spanning trees in $G$ is equal to any…

We define the Laplacian matrix and the Jacobian group of a finite graph of groups. We prove analogues of the matrix tree theorem and the class number formula for the order of the Jacobian of a graph of groups. Given a group $G$ acting on a…

By revisiting the Kirchhoff's Matrix-Tree Theorem, we give an exact formula for the number of spanning trees of a graph in terms of the quantum relative entropy between the maximally mixed state and another state specifically obtained from…

A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We…

We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…

A depth-first search version of Dhar's burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the…

The Matrix-Tree Theorem states that the number of spanning trees of a graph is given by the absolute value of any cofactor of the Laplacian matrix of the graph. We propose a very short proof of this result which amounts to comparing Taylor…

We study the problem of maximizing the number of spanning trees in a connected graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm…

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r >= 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from…

We present a new characterization of $k$-trees based on their reduced clique graphs and $(k+1)$-line graphs, which are block graphs. We explore structural properties of these two classes, showing that the number of clique-trees of a…

Fast exact algorithms are known for Hamiltonian paths in undirected and directed bipartite graphs through elegant though involved algorithms that are quite different from each other. We devise algorithms that are simple and similar to each…

Chip firing provides a way to study the sandpile group (also known as the Jacobian) of a graph. We use a generalized version of chip firing to bound the number of invariant factors of the critical group of an arithmetical structure on a…

Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This…

The number of rooted spanning forests divided by the number of spanning rooted trees in a graph G with Kirchhoff matrix K is the spectral quantity tau(G)= det(1+K)/det(K) of G by the matrix tree and matrix forest theorems. We prove that…