Related papers: Effective resistance and spanning trees in complet…

We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs.…

We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson…

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices…

With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The…

A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end…

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 study a new type of random minimum spanning trees. It is built on the complete graph where each vertex is given a weight, which is a positive real number. Then, each edge is given a capacity which is a random variable that only depends…

Given a connected undirected graph G = [V; E] where |E| =2(|V| -1), we present two algorithms to check if G can be decomposed into two edge disjoint spanning trees, and provide such a decomposition when it exists. Unlike previous algorithms…

Resistance distance computation is a fundamental problem in graph analysis, yet existing random walk-based methods are limited to approximate solutions and suffer from poor efficiency on small-treewidth graphs (e.g., road networks). In…

Graph operations or products play an important role in complex networks. In this paper, we study the properties of $q$-subdivision graphs, which have been applied to model complex networks. For a simple connected graph $G$, its…

Spanning trees $T_1,T_2, \dots,T_k$ of $G$ are $k$ completely independent spanning trees if, for any two vertices $u,v\in V(G)$, the paths from $u$ to $v$ in these $k$ trees are pairwise edge-disjoint and internal vertex-disjoint. Hasunuma…

Any graph can be considered as a network of resistors, each of which has a resistance of $1 \Omega.$ The resistance distance $r_{ij}$ between a pair of vertices $i$ and $j$ in a graph is defined as the effective resistance between $i$ and…

We study the problem of distance-preserving graph compression for weighted paths and trees. The problem entails a weighted graph $G = (V, E)$ with non-negative weights, and a subset of edges $E^{\prime} \subset E$ which needs to be removed…

Let $\mathcal{G}_{n,r,s}$ denote a uniformly random $r$-regular $s$-uniform hypergraph on the vertex set $\{1,2,\ldots, n\}$. We establish a threshold result for the existence of a spanning tree in $\mathcal{G}_{n,r,s}$, restricting to $n$…

Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The resistance distance $R_G(x,y)$ between two vertices $x,y$ of $G$ is defined to be the effective resistance between the two vertices in the corresponding…

Using the method of spectral decimation and a modified version of Kirchhoffs Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely…

Our previous paper shows that the (vertex) spanning tree degree enumerator polynomial of a connected graph $G$ is a real stable polynomial (id est is non-zero if all variables have positive imaginary parts) if and only if $G$ is…

In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…

In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let $(X,Y)$ be the bipartition of the complete bipartite graph $K_{m,n}$ with…

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the…