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Register a new userThe subdivision of large simplicial cones in Normaliz
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Normaliz is an open-source software for the computation of lattice points in rational polyhedra, or, in a different language, the solutions of linear diophantine systems. The two main computational goals are (i) finding a system of generators of the set of lattice points and (ii) counting elements degree-wise in a generating function, the Hilbert Series. In the homogeneous case, in which the polyhedron is a cone, the set of generators is the Hilbert basis of the intersection of the cone and the lattice, an affine monoid. We will present some improvements to the Normaliz algorithm by subdividing simplicial cones with huge volumes. In the first approach the subdivision points are found by integer programming techniques. For this purpose we interface to the integer programming solver SCIP to our software. In the second approach we try to find good subdivision points in an approximating overcone that is faster to compute.
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In this article we describe mathematically relevant extensions to Normaliz that were added to it during the support by the DFG SPP Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie: nonpointed cones, rational polyhedra, homogeneous systems of parameters, bottom decomposition, class groups and systems of module generators of integral closures.
The subdivision graph $mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The emph{subdivision-vertex corona} of $G_1$ and $G_2$, denoted by $G_1odot G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $|V(G_1)|$ copies of $G_2$, all vertex-disjoint, by joining the $i$th vertex of $V(G_1)$ to every vertex in the $i$th copy of $G_2$. The emph{subdivision-edge corona} of $G_1$ and $G_2$, denoted by $G_1circleddash G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $|I(G_1)|$ copies of $G_2$, all vertex-disjoint, by joining the $i$th vertex of $I(G_1)$ to every vertex in the $i$th copy of $G_2$, where $I(G_1)$ is the set of inserted vertices of $mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1odot G_2$ (respectively, $G_1circleddash G_2$) in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, the results on the spectra of $G_1odot G_2$ (respectively, $G_1circleddash G_2$) enable us to construct infinitely many pairs of cospectral graphs. The adjacency spectra of $G_1odot G_2$ (respectively, $G_1circleddash G_2$) help us to construct many infinite families of integral graphs. By using the Laplacian spectra, we also obtain the number of spanning trees and Kirchhoff index of $G_1odot G_2$ and $G_1circleddash G_2$, respectively.
We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$s density. This generalizes a well-known classical result of ErdH{o}s and Gallai cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.
Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{j}$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let $mu_{j}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $mu_{k}(X)$ for $kgeq d$ and $mu_{d-1}(X)$. In particular, we establish the following vanishing result: If $mu_{d-1}(X)>(1-binom{k+1}{d}^{-1})n$, then $tilde{H}^{j}(X;mathbb{R})=0$ for all $d-1leq j leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martinez-Sandoval and Montejano for general position sets in matroids.
The subdivision graph $mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1dot{vee}G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$. The emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by $G_1underline{vee}G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, where $I(G_1)$ is the set of inserted vertices of $mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1dot{vee}G_2$ (respectively, $G_1underline{vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$, in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, these results enable us to construct infinitely many pairs of cospectral graphs. We also give the number of the spanning trees and the Kirchhoff index of $G_1dot{vee}G_2$ (respectively, $G_1underline{vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$.
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