Related papers: The counting matrix of a simplicial complex

A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. This matrix is always unimodular so that the inverse g of L has integer entries g(x,y). In…

For a simplicial complex with n sets, let W^-(x) be the set of sets in G contained in x and W^+(x) the set of sets in G containing x. An integer-valued function h on G defines for every A subset G an energy E[A]=sum_x in A h(x). The…

If h is a ring-valued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green…

We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite abstract simplicial complex G matches the number b of even dimensional simplices in G and that the number n of negative eigenvalues matches…

We prove a Gauss-Bonnet formula X(G) = sum_x K(x), where K(x)=(-1)^dim(x) (1-X(S(x))) is a curvature of a vertex x with unit sphere S(x) in the Barycentric refinement G1 of a simplicial complex G. K(x) is dual to (-1)^dim(x) for which…

The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As…

We look at connection Laplacians L,g defined by a field h:G to K, where G is a finite set of sets and K is a normed division ring which does not need to be commutative, nor associative but has a conjugation leading to the norm as the square…

Given an abstract simplicial complex G, the connection graph G' of G has as vertex set the faces of the complex and connects two if they intersect. If A is the adjacency matrix of that connection graph, we prove that the Fredholm…

We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G…

The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the eigenvalues for subcomplexes. This gives general upper bounds…

While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu characteristics w_k(G) = sum_(x_1,x_2,...,x_k) w(x_1)...w(x_k) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial…

Classical simplicial cohomology on a simplicial complex G deals with functions on simplices x in G. Quadratic cohomology deals with functions on pairs of simplices (x,y) in G x G that intersect. If K,U is a closed-open pair in G, we prove…

Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ...), and generating function f(x) = sum(k=1 v(k-1) x^k = v0 x + v1 x^2+ v2 x^3 + ..., the Euler characteristic of G can be written as chi(G)=f(0)-f(-1). We study here…

Let $G$ be a finite abelian group of order $n$ and let $\Delta_{n-1}$ denote the $(n-1)$-simplex on the vertex set $G$. The sum complex $X_{A,k}$ associated to a subset $A \subset G$ and $k < n$, is the $k$-dimensional simplicial complex…

We define the set of almost-intertwining matrices to be all triples(X,Y,Z) of n x n matrices for which XZ=YX+T for some rank one matrix T. A surprisingly simple formula is given for tau-functions of the KP hierarchy in terms of such…

We define a ring R of geometric objects G generated by finite abstract simplicial complexes. To every G belongs Hodge Laplacian H as the square of the Dirac operator determining its cohomology and a unimodular connection matrix L). The sum…

Let $G$ be a graph with the vertex set $ \lbrace v_1,\ldots,v_n \rbrace$. The Seidel matrix of $G$ is an $n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $ v_{i} $ and $ v_{j} $ are adjacent and otherwise is $ 1…

If $M$ is a set of nonsingular $k\times k$ matrices then for many pairs of matrices, $A,B\in M,$ the sum is nonsingular, $\det(A+B)\neq 0.$ We prove a more general statement on nonsingular sums with an application.

For a real reductive linear Lie group G, the space of Whittaker functions is the representation space induced from a non-degenerate unitary character of the Iwasawa nilpotent subgroup. Defined are the standard Whittaker (g,K)-modules, which…

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G', connecting two simplices if they intersect. We…