Related papers: Linear Algebra and Number of Spanning Trees

We present a determinantal formula for the number of spanning trees of a complete multipartite graph containing a given spanning forest $F$. Our approach relies on the Generalized Matrix Determinant Lemma and Jacobi's formula for the…

A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree…

Counting the number of spanning trees in specific classes of graphs has attracted increasing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained in the 2020s. The main method is…

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the…

In deep neural networks, better results can often be obtained by increasing the complexity of previously developed basic models. However, it is unclear whether there is a way to boost performance by decreasing the complexity of such models.…

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…

We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$,…

This paper investigates some relationship between the algebraic connectivity and the clique number of graphs. We characterize all extremal graphs which have the maximum and minimum the algebraic connectivity among all graphs of order $n$…

Graphs are fundamental data structures which concisely capture the relational structure in many important real-world domains, such as knowledge graphs, physical and social interactions, language, and chemistry. Here we introduce a powerful…

The network topology can be described by the number of nodes and the interconnections among them. The degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability…

Degree correlation is an important topological property common to many real-world networks. In this paper, the statistical measures for characterizing the degree correlation in networks are investigated analytically. We give an exact proof…

In finite group theory, studying the prime graph of a group has been an important topic for almost the past half-century. Recently, prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the…

In this paper we examine the classes of graphs whose $K_n$-complements are trees and quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph $H$ of $K_n$, the $K_n$-complement of $H$ is the graph…

We present a new framework to measure the intrinsic properties of (deep) neural networks. While we focus on convolutional networks, our framework can be extrapolated to any network architecture. In particular, we evaluate two network…

The spanning tree heuristic is a commonly adopted procedure in network inference and estimation. It allows one to generalize an inference method developed for trees, which is usually based on a statistically rigorous approach, to a…

We study the height of a spanning tree $T$ of a graph $G$ obtained by starting with a single vertex of $G$ and repeatedly selecting, uniformly at random, an edge of $G$ with exactly one endpoint in $T$ and adding this edge to $T$.

We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed…

In this paper, we use a partition of the links of a network in order to uncover its community structure. This approach allows for communities to overlap at nodes, so that nodes may be in more than one community. We do this by making a node…

We study probability distributions over free algebras of trees. Probability distributions can be seen as particular (formal power) tree series [Berstel et al 82, Esik et al 03], i.e. mappings from trees to a semiring K . A widely studied…