相关论文: The Forest Metrics for Graph Vertices

We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph G is a spanning subgraph of G that consists of disjoint diverging trees and has the maximum possible number of…

In this paper we offer a metric similar to graph edit distance which measures the distance between two (possibly infinite)weighted graphs with finite norm (we define the norm of a graph as the sum of absolute values of its edges). The main…

There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…

Using the matrix-forest theorem and the Parisi-Sourlas trick we formulate and solve a one-matrix model with non-polynomial potential which provides perturbation theory for massive spinless fermions on dynamical planar graphs. This is a…

Let $G$ be a connected graph on $n$ vertices and $d_{ij}$ be the length of the shortest path between vertices $i$ and $j$ in $G$. We set $d_{ii}=0$ for every vertex $i$ in $G$. The squared distance matrix $\Delta(G)$ of $G$ is the $n\times…

The graph is one of the most widely used mathematical structures in engineering and science because of its representational power and inherent ability to demonstrate the relationship between objects. The objective of this work is to…

For a graph $G=(V, E)$ and $i, j\in V$, denote the distance between $i$ and $j$ in $G$ by $D(i, j)$ and the degrees of $i$, $j$ by $d_i$, $d_j$, respectively. Let $f(D(i, j), d_{i}, d_{j})$ be a function symmetric in $i$ and $j$. Define a…

We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…

Vertex similarity is a major problem in network science with a wide range of applications. In this work we provide novel perspectives on finding (dis)similar vertices within a network and across two networks with the same number of vertices…

We study the inertia of distance matrices of weighted graphs. Our novel congruence-based proof of the inertia of weighted trees extends to a proof for the inertia of weighted unicyclic graphs whose cycle is a triangle. Partial results are…

This work briefly explores the possibility of approximating spatial distance (alternatively, similarity) between data points using the Isolation Forest method envisioned for outlier detection. The logic is similar to that of isolation: the…

Simplifying graphs is a very applicable problem in numerous domains, especially in computational geometry. Given a geometric graph and a threshold, the minimum-complexity graph simplification asks for computing an alternative graph of…

In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of…

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…

We consider the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$. We call $G_n$ the straight linear 2-tree on $n$ vertices. Using $\Delta$--Y transformations and identities…

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…

Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph $G$ that can be completely unrelated to the metric. The aim is to partition the $n$…

Distance well-defined graphs consist of connected undirected graphs, strongly connected directed graphs and strongly connected mixed graphs. Let $G$ be a distance well-defined graph, and let ${\sf D}(G)$ be the distance matrix of $G$.…

Finding a new mathematical representations for graph, which allows direct comparison between different graph structures, is an open-ended research direction. Having such a representation is the first prerequisite for a variety of machine…

The forest matrix plays a crucial role in network science, opinion dynamics, and machine learning, offering deep insights into the structure of and dynamics on networks. In this paper, we study the problem of querying entries of the forest…