Related papers: Matrix-Forest Theorems

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance…

A (pseudo-)metric $D$ on a finite set $X$ is said to be a `tree metric' if there is a finite tree with leaf set $X$ and non-negative edge weights so that, for all $x,y \in X$, $D(x,y)$ is the path distance in the tree between $x$ and $y$.…

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, \chi)$ where $T$ is a tree and $\chi : V(T) \rightarrow 2^{V(G)}$ is a function satisfying the following two axioms:…

In this paper, we relate the seemingly unrelated concepts of treewidth and boxicity. Our main result is that, for any graph G, boxicity(G) <= treewidth(G) + 2. We also show that this upper bound is (almost) tight. Our result leads to…

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\textit{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\textit{weakly even}$ if every leaf…

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S\subseteq V(G)$, $\omega(G\setminus S)<\sum_{v\in S}(f(v)-2)+2+\omega(G[S])$, then $G$ has a spanning tree $T$…

Let $G$ be a simple graph, $A(G)$ its adjacency matrix, and $D(G)$ its diagonal degree matrix. In 2022, \citeauthor{Wang2020} (\cite{Wang2020}) defined the family of matrices $L_\alpha$ as the convex linear combination: \[ L_\alpha(G) =…

Let $M=(m_{ij})$ be a symmetric matrix of order $n$ whose elements lie in an arbitrary field $\mathbb{F}$, and let $G$ be the graph with vertex set $\{1,\ldots,n\}$ such that distinct vertices $i$ and $j$ are adjacent if and only if $m_{ij}…

It is well known that if $G = (V, E)$} is a multigraph and $X\subset V$ is a subset of even order, then $G$ contains a spanning forest $H$ such that each vertex from $X$ has an odd degree in $H$ and all the other vertices have an even…

Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be…

There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such…

The arboricity $\Gamma(G)$ of an undirected graph $G =(V,E)$ is the minimal number $k$ such that $E$ can be partitioned into $k$ forests on $V$. Nash-Williams' formula states that $k = \lceil \gamma(G) \rceil$, where $\gamma(G)$ is the…

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of…

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…

Let $(G,w)$ be an undirected weighted graph. The group inverse of $(G,w)$ is the weighted graph with the adjacency matrix $A^{\#}$, where $A$ is the adjacency matrix of $(G,w)$. We study the group inverse of singular weighted trees. It is…

The induced matching width of a tree decomposition of a graph $G$ is the cardinality of a largest induced matching $M$ of $G$, such that there exists a bag that intersects every edge in $M$. The induced matching treewidth of a graph $G$,…

The energy of a graph is defined as the sum the absolute values of the eigenvalues of its adjacency matrix. A graph G on n vertices is said to be borderenergetic if its energy equals the energy of the complete graph Kn. In this paper, we…

The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…

Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length…

A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$…