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 cofactor of an entry of $L(G)$. A rooted forest is a union of disjoint rooted trees. We consider the matrix $W(G)=I+L(G)$ and prove that the $(i,j)$-cofactor of $W(G)$ is equal to the number of spanning rooted forests of $G$, in which the vertices $i$ and $j$ belong to the same tree rooted at $i$. The determinant of $W(G)$ equals the total number of spanning rooted forests, therefore the $(i,j)$-entry of the matrix $W^{-1}(G)$ can be considered as a measure of relative ''forest-accessibility'' of vertex $i$ from $j$ (or $j$ from $i$). These results follow from somewhat more general theorems we prove, which concern weighted multigraphs. The analogous theorems for (multi)digraphs are also established. These results provide a graph-theoretic interpretation for the adjugate to the Laplacian characteristic matrix.
We study spanning diverging forests of a digraph and related matrices. It is shown that the normalized matrix of out forests of a digraph coincides with the transition matrix in a specific observation model for Markov chains related to the digraph. Expression are given for the Moore-Penrose generalized inverse and the group inverse of the Kirchhoff (Laplacian) matrix. These expressions involve the matrix of maximum out forest of the digraph. Every matrix of out forests with a fixed number of arcs and the normalized matrix of out forests are represented as polynomials in the Kirchhoff matrix; w…
A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra. We give here two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both our proofs yield polynom…
In this paper, we study the graph polynomial that counts spanning rooted forests f_g of a given graph. This polynomial has a remarkable reciprocity property. We give a new bijective proof for this theorem which has Prufer coding as a special case.
We study the matrices Q_k of in-forests of a weighted digraph G and their connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of L. The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the L…
A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal--Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented. In addition, it will be shown that a quasi-forest is shellable if and only if its $h$-vector $(h_0, h_1, h_2, ...)$ satisfies $h_i = 0$ for $i > 1$.
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 arcs. If a digraph contains any out arborescences, then maximum out forests coincide with them. We provide a new proof to the Markov chain tree theorem saying that the matrix of Ces`aro limiting probabilities of an arbitrary stationary finite Markov chain coincides with the normalized matrix of maximum out forests of the weighted digraph that corresponds to…
We propose a family of graph structural indices related to the Matrix-forest theorem. The properties of the basic index that expresses the mutual connectivity of two vertices are studied in detail. The derivative indices that measure "dissociation," "solitariness," and "provinciality" of vertices are also considered. A nonstandard metric on the set of vertices is introduced, which is determined by their connectivity. The application of these indices in sociometry is discussed.
For a graph G, let f_{ij} be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_{ij}'s can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The {\em doubly stochastic graph matrix} is the matrix F=(f_{ij})/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^{-1}=I+L, where L is the Laplacian matrix of G. We show that fo…
The matrices of spanning rooted forests are studied as a tool for analysing the structure of digraphs and measuring their characteristics. The problems of revealing the basis bicomponents, measuring vertex proximity, and ranking from preference relations / sports competitions are considered. It is shown that the vertex accessibility measure based on spanning forests has a number of desirable properties. An interpretation for the normalized matrix of out-forests in terms of information dissemination is given. Keywords: Laplacian matrix, spanning forest, matrix-forest theorem, proximity measure,…
The Erdős-Gallai Theorem states that every graph of average degree more than $l-2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erdős-Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=(\bigcup_{i=1}^kP_{2a_i})\bigcup(\bigcup_{i=1}^lP_{2b_i+1})$ be $k+l$ disjoint paths of order $2a_1, \ldots, 2a_{k}, 2b_1+1, \ldots, 2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\geq 2$. If the minimum degree $δ(G)\ge \sum_{i=1}^ka_i+\sum_{i=1}^lb_i-1$, then $F\subseteq G$ except several classes of graphs …
We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij} is the (i,j)-entry of the {\em relative forest accessibility matrix} Q(ε)=(I+εL)^{-1}, L is the Laplacian matrix of the (weighted) (multi)graph, and εis a positive parameter. By the matrix-forest theorem, the (i,j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to …
We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these determine the tree and the $d$ length of all edges. Here we consider distortions $\d$ of $d$ such that for all leaves $u$ and $v$ it holds that $|d(u,v) - \d(u,v)| < f/2$ if either $d(u,v) < M$ or $\d(u,v) < M$, where $d$ satisfies $f \leq d(e) \leq g$ for all edges $e$. Given such distortions we show how to reconstruct in polynomial time a forest $T_1,...,…
The Lewis Carroll Identity expresses the determinant of a matrix in terms of subdeterminants obtained by deleting one row and column or a pair of rows and columns. Using the matrix tree theorem, we can convert this into an equivalent identity involving sums over pairs of forests. Unlike the Lewis Carroll Identity, the Forest Identity involves no minus signs. In 2011, Vlasev and Yeats suggested that such a Forest Identity could be proven using edge transfers similar to Zeilberger's 1997 matrix proof. However, until now, such an algorithm has not yet been developed. In this paper, we provide thi…
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 that under Barycentric refinements, the tree index T(G)=log(det(K))/n and forest index F(G)=log(det(1+K))/n and so the tree-forest index i=F-G=log(tau(G))/n converge to numbers that only depend on the size of the maximal clique in the graph. In the 1-dimensional case, all numbers are known: T(G)=0, F(G)=i(G) =2 log(phi), where phi is the golden ratio. The conve…
Schmid College of Science and Technology, Chapman University（查普曼大学施密德科学与技术学院）；Institute of Quantum Studies, Chapman University（查普曼大学量子研究学院）；Department of Physics and Astronomy, Clemson University（克莱姆森大学物理与天文学系）
Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_α(G)$ of $A (G)$ and $D (G) $ by $$A_α(G)=αD(G)+(1-α)A(G)$$ for any real number $0\leqα\leq1$. The \emph{$α$-index} of $G$ is the largest eigenvalue of $A_α(G)$. In this paper, we determine the maximum $α$-index and characterize all extremal graphs for $K_r$ minor-free graphs, $K_{s,t}$ minor-free graphs, and star-forest-free graphs for any $0<α<1$ by unified eigenvector approach, respectively.
A matrix is called acyclic if replacing the diagonal entries with $0$, and the nonzero diagonal entries with $1$, yields the adjacency matrix of a forest. In this paper we show that null space and the rank of a acyclic matrix with $0$ in the diagonal is obtained from the null space and the rank of the adjacency matrix of the forest by multipliying by non-singular diagonal matrices. We combine these methods with an algorithm for finding a sparsest basis of the null space of a forest to provide an optimal time algorithm for finding a sparsest basis of the null space of acyclic matrices with $0$ …
The Dodgson/Muir Identity is an identity on determinants of matrix minors that generalizes the Dodgson Identity. Using the matrix tree theorem, we create an equivalent forest identity on ordered sets of $k$ forests. We then prove the generalized Forest Identity, and by extension the Dodgson/Muir Identity, using an edge-swapping involution. This algorithm is a generalization of the Red Hot Potato algorithm, developed by the author in 2021 to prove the Dodgson Identity.
Let $\mathcal{F}$ be a family of graphs. The Turán number $ex(n;\mathcal{F})$ is defined to be the maximum number of edges in a graph of order $n$ that is $\mathcal{F}$-free. In 1959, Erdős and Gallai determined the Turán number of $M_{k+1}$ (a matching of size $k+1$) as follows: \[ ex(n;M_{k+1})= \max\left\{\binom{2k+1}{2},\binom{n}{2}-\binom{n-k}{2}\right\}. \] Since then, there has been a lot of research on Turán number of linear forests. A linear forest is a graph whose connected components are all paths or isolated vertices. Let $\mathcal{L}_{n,k}$ be the family of all linear forests of o…
For a poset whose Hasse diagram is a rooted plane forest $F$, we consider the corresponding tree descent polynomial $A_F(q)$, which is a generating function of the number of descents of the labelings of $F$. When the forest is a path, $A_F(q)$ specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of $A_F(q)$ is unimodal and that if $\{T_{n}\}$ is a sequence of trees with $|T_{n}| = n$ and maximal down degree $D_{n} = O(n^{0.5-ε})$ then the number of descents in a labeling of $T_{n}$ is asymptotically normal.
A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set if it is a maximum stable set of the subgraph of G spanned by the union of S and N(S), where N(S) is the neighborhood of S. One theorem of Nemhauser and Trotter Jr., working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for forests. Namely, we show that for any non-empty local maximum sta…
Institute of Control Sciences of the Russian Academy of Sciences（俄罗斯科学院控制科学研究所）
The matrices of spanning rooted forests are studied as a tool for analysing the structure of networks and measuring their properties. The problems of revealing the basic bicomponents, measuring vertex proximity, and ranking from preference relations / sports competitions are considered. It is shown that the vertex accessibility measure based on spanning forests has a number of desirable properties. An interpretation for the stochastic matrix of out-forests in terms of information dissemination is given.
In 2001, Brualdi and Hollingsworth proved that an edge-colored balanced complete bipartite graph Kn,n with a color set C = {1,2,3,..., 2n-1} has a heterochromatic spanning tree if the number of edges colored with colors in R is more than |R|^2 /4 for any non-empty subset R \subseteq C, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors, namely, any color appears at most once. In 2010, Suzuki generalized heterochromatic graphs to f-chromatic graphs, where any color c appears at most f(c). Moreover, he presented a necessary and sufficient condition for grap…
The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of these random permutations. The main results show that the distributions of these numbers are asymptotically normal. The proof uses representations of random tree and forest permutations that enable us to express the number of occurrences of a pattern by a type of $U$-statistics; we then use general limit theorems for the latter.
Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Turán number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study of Turán numbers is a central problem in graph theory. The goal of this paper is to generalize a theorem of Lidický, Liu and Palmer [{\it Electron.\ J.\ of Combin.}\ {\bf 20} (2016)] that determines $\mathrm{ex}(n,F)$ for $F$ a forest of stars. In particular, we consider generalizations of the problem to three different well-studied hypergraph settings and in…
The All Minors Matrix Tree Theorem states that the determinant of any submatrix of a matrix whose columns sum to zero can be computed as a sum over certain oriented forests. We offer a particularly short proof of this result, which amounts to comparing Taylor series expansions.
For a loopless multigraph $G$, the fractional arboricity $Arb(G)$ is the maximum of $\frac{|E(H)|}{|V(H)|-1}$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G)\le k+\frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture was previously proved for $d=k+1$ and for $k=1$ when $d \le 6$. We prove it for all $d$ when $k \le 2$, except for $(k,d)=(2,1)$.
We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with $k$ edges are $\# W[1]$-hard when parameterized by $k$. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for $k$-forests implies $\# W[1]$-hardness of the problem of counting bases of a matroid when parameterized by rank or nullity, even if the matroid is restricted to be representable over a field of characteristic $2$. We c…
A Laplacian matrix is a square real matrix with nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with the absolute values of the off-diagonal entries not exceeding 1/n, where n is the order of the matrix. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices are semiconvergent. The multiplicities of 0 and 1 as the eigenvalu…
The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher's result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumer…
Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Turán number of $\mathcal{F}$ is the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,\mathcal{F})$. When $T = K_r$, $ex(n, K_r, \mathcal{F})$ is a function specifying the maximum possible number of $r$-cliques in an $\mathcal{F}$-free graph on $n$ vertices. A linear forest is a forest whose connected components are all paths and isolated vertices. Let $\mathcal{L}_{k}$ be the family of all linear forests of size $k$ without isolated vertices. In this paper, we obtained the…
Let $m_{ij}$ be the mean first passage time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. Let $G$ be the weighted digraph without loops whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. We give a graph-theoretic interpretation to $m_{ij}$. Namely, We show that $m_{ij}=f_{ij}/q_j$ if $i\ne j$ and $m_{ij}=1/\tilde q_j$ if $i=j$, where $f_{ij}$ is the total weight of 2-tree spanning converging forests in $G$ that have one tree containing $i$ and t…
A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices $u$ and $v$ in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating $u$ and $v$ divided by the number of spanning trees in the graph. There are also well…
A graph $G$ is $\ell$-hamiltonian if for any linear forest $F$ of $G$ with $\ell$ edges, $F$ can be extended to a hamiltonian cycle of $G$. We give a sharp upper bound for the maximum number of cliques of a fixed size in a non-$\ell$-hamiltonian graph. Furthermore, we prove stability for the bound: if a non-$\ell$-hamiltonian graph contains almost the maximum number of cliques, then it must be a subgraph of one of two examples.
We prove a generalization of Kirchhoff's matrix-tree theorem in which a large class of combinatorial objects are represented by non-Gaussian Grassmann integrals. As a special case, we show that unrooted spanning forests, which arise as a q \to 0 limit of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. We show that this latter model can be mapped, to all orders in perturbation theory, onto the N-vector model at N=-1 or, equivalently, onto the sigma-model taking values in the unit supersphere in R^{1|2}. It follows t…
Let $A$ be an acyclic symmetric matrix of order $n$. There is a weighted forest $F$ whose adjacency matrix is $A$. In this paper, using some results on matching polynomials, we provide an explicit formula for eigenvectors of $A$.
We prove a common generalization of the celebrated Sauer-Spencer packing theorem and a theorem of Brandt concerning finding a copy of a tree inside a graph. This proof leads to the characterization of the extremal graphs in the case of Brandt's theorem: If $G$ is a graph and $F$ is a forest, both on $n$ vertices, and $3Δ(G)+\ell^*(F)\leq n$, then $G$ and $F$ pack unless $n$ is even, $G=\frac{n}{2}K_2$ and $F=K_{1,n-1}$; where $\ell^*(F)$ is the difference between the number of leaves and twice the number of nontrivial components of $F$.
We calculate determinants of weighted sums of reflections and of (nested) commutators of reflections. The results obtained generalize the Kirchhoff's matrix-tree theorem and the matrix-3-hypertree theorem by G.\,Massbaum and A.\,Vaintrob.
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 expansions.
In this short note, we revisit Zeilberger's proof of the classical matrix-tree theorem and give a unified concise proof of variants of this theorem, some known and some new.
Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system $D_n$ (the classical theorem corresponds to the $A_n$-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest.
Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph $G$ and the dimer model defined on a decorated version $\GD$ of this graph \cite{Fisher}. In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristi…
The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q -> 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n -> -1, the expansion parameter t counting the number of components in the forest. We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k = 3, a correspondence is found with the Kostov solution of the loop-…
