Related papers: Higher determinants and the matrix-tree theorem

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…

A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an…

The image of the principal minor map for n x n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial…

We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair.…

Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. Recently an analogue was obtained for quantum graphs relating the number of spanning trees…

We prove a formula for a characteristic polynomial of an operator expressed as a polynomial of rank 1 operators. The formula uses a discrete analog of path integration and implies a generalization of the Forman-Kenyon's formula [4,6] for a…

In 1987 Hiroshi Maehara conjectured that a graph can be represented by vectors considered adjacent when not orthogonal (a faithful orthogonal representation) in codimension the minimum degree of the graph. Without settling the conjecture,…

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an $r$-tree if, additionally, it is $r$-uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree $T$ with…

The notion of directed treewidth was introduced by Johnson, Robertson, Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as a first step towards an algorithmic metatheory for digraphs. They showed that some…

We define two families of determinantal random spanning subgraphs of a finite connected graph, one supported by acyclic spanning subgraphs (spanning forests) with fixed number of connected components, the other by connected spanning…

Lorea [11] and later Frank et al. [8] generalized graphic matroids to hypergraphic matroids. In [8], the authors introduced hypertrees as a generalization of spanning trees and proved a form of the theorem of Tutte [18] and Nash-Williams…

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…

One of the aims of this paper is to provide a short survey on the Z2-graded, the symmetric and the left (right) generalizations of the classical determinant theory for square matrices with entries in an arbitrary (possibly non-commutative)…

We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and trees on $n+1$ vertices where $k$ children of the…

The Tutte polynomial of a connected graph was originally defined by Tutte as a sum over all spanning trees of monomials depending on a fixed linear order on the set of edges. Tuttle proved that while these monomials do depend on the linear…

We consider the number of spanning trees in circulant graphs of $\beta n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $\beta n$ factors, while we derive a formula of $\beta-1$…

The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-$k$ Steiner distance hypermatrix of an $n$-vertex graph is the $n \times \cdots \times n$ ($k$…

We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-$k$ matrices. Specifically, we prove that if $X_1,\dots ,X_n$ are…

In [Directed tree-width, J. Combin. Theory Ser. B 82 (2001), 138-154] we introduced the notion of tree-width of directed graphs and presented a conjecture, formulated during discussions with Noga Alon and Bruce Reed, stating that a digraph…

We make progress towards characterizing the algebraic matroid of the determinantal variety defined by the minors of fixed size of a matrix of variables. Our main result is a novel family of base sets of the matroid, which characterizes the…