Related papers: Higher determinants and the matrix-tree theorem
George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values…
We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \cite{Lambek} and Johnstone \cite{Johnstone} used for monoid and category emedding results. We show that they are planar…
Courcelle's Theorem states that on graphs $G$ of tree-width at most $k$ with a given tree-decomposition of size $t(G)$, graph properties $\mathcal{P}$ definable in Monadic Second Order Logic can be checked in linear time in the size of…
We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the…
The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of $\aleph_n$ is $n+1$. We record a new proof and mild strengthening of this theorem; our more fundamental aim, though, is some…
In this paper, we address the Ehrenborg's conjecture which proposes that for any bipartite graph the number of spanning trees does not exceed the product of the degrees of the vertices divided by the product of the sizes of the graph…
A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider…
Muller and Schupp introduced the concept of context-free graphs (originating from Cayley graphs of context-free groups). These graphs are always tree-like (i.e. quasi-isometric to a tree) and in this paper we investigate the subclass of…
The N-tree approximation scheme, introduced in the context of random directed polymers, is here applied to the computation of the maximum Lyapunov exponent in a coupled map lattice. We discuss both an exact implementation for small…
We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Z_k of order k. We then demonstrate how such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn…
Let $\mathcal{H}$ be a $k$-uniform hypergraph. A chain in $\mathcal{H}$ is a sequence of its vertices such that every $k$ consecutive vertices form an edge. In 1999 Katona and Kierstead suggested to use chains in hypergraphs as the…
Antimatroids were discovered by Dilworth in the context of lattices [4] and introduced by Edelman and Jamison as convex geometries in[5]. The author of the current paper independently discovered (possibly infinite) antimatroids in the…
We prove `twisted' versions of Kirchhoff's network theorem and Kirchhoff's matrix-tree theorem on connected finite graphs. Twisting here refers to chains with coefficients in a flat unitary line bundle.
The objective of this note is to establish the Determinant Majorization Formula $F(A)^{1\over N} \geq \det(A)^{1\over n}$ for all operators $F$ determined by an invariant Garding-Dirichlet polynomial of degree $N$ on symmetric $n \times n$…
Given a graph G, an incidence matrix N(G) is defined for the set of distinct isomorphism types of induced subgraphs of G. If Ulam's conjecture is true, then every graph invariant must be reconstructible from this matrix, even when the…
In this paper we give the absolutely new proof of a conjecture of R.F.Scott(1881) on the permanent of a Cauchy matrix $\ls \frac{1}{x_i-y_j} \rs_{1 \leqslant i,j \leqslant n},$ where $x_1, ..., x_n$ and $y_1, ..., y_n$ are the distinct…
Transfer Maps, sometimes called norm maps, for Milnor's $K$-theory were first defined by Bass and Tate (1972) for simple extensions of fields via tame symbol and Weil's reciprocity law, but their functoriality had not been settled until…
A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let…
Let k be a local field and let A be the two-by-two matrix algebra over k. In our previous work we developed a theory that allows the computation of the set of maximal orders in A containing a given suborder. This set is given as a sub-tree…
Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single vertex. We prove a large deviation principle in…