Related papers: A New Matrix-Tree Theorem
We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that…
The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph.…
We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $\gamma$-polynomials of their associated graph-associahedra. The partial…
It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4+2 leaves and that this can be improved to (n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize the second…
Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral…
We prove that every 2-sphere graph different from a prism can be vertex 4-colored in such a way that all Kempe chains are forests. This implies the following three tree theorem: the arboricity of a discrete 2-sphere is 3. Moreover, the…
This paper formulates a novel problem on graphs: find the minimal subset of edges in a fully connected graph, such that the resulting graph contains all spanning trees for a set of specifed sub-graphs. This formulation is motivated by an…
We present an elementary proof of a generalization of Kirchoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which only involves the…
We study the computation of our recently introduced Whitney polynomial and the enumeration of the spanning hypertrees for hypermaps whose hyperedges have length at most $3$. This is a class of hypermaps where the computation of the above…
Restrictions of incidence-preserving path maps produce an oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of…
A tree is called k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. In this paper we prove that every 3-regular connected graph with n vertices such that n is greater than 8 has spanning sub tree with at most…
In 1966, Cummins introduced the "tree graph": the tree graph $\mathbf{T}(G)$ of a graph $G$ (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge,…
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs…
We present a version of the matrix-tree theorem, which relates the determinant of a matrix to sums of weights of arborescences of its directed graph representation. Our treatment allows for non-zero column sums in the parent matrix by…
A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…
The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper,…
In this paper, we find recursive formulas for the Tutte polynomial of a family of small-world networks: Farey graphs, which are modular and have an exponential degree hierarchy. Then, making use of these formulas, we determine the number of…
By the classical Sturm's theorem, the number of distinct real roots of a given real polynomial $f(x)$ within any interval $(a,b]$ can be expressed by the number of variations in the sign of the Sturm chain at the bounds. Through…
The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs in open. The complexities of recognising many subclasses of SOGs are known.…