Index theorems for quantum graphs

In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a "Laplacian") with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H =A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.Comment: 19 pages, Institute of Physics LaTe

Graphulo Implementation of Server-Side Sparse Matrix Multiply in the Accumulo Database

The Apache Accumulo database excels at distributed storage and indexing and is ideally suited for storing graph data. Many big data analytics compute on graph data and persist their results back to the database. These graph calculations are often best performed inside the database server. The GraphBLAS standard provides a compact and efficient basis for a wide range of graph applications through a small number of sparse matrix operations. In this article, we implement GraphBLAS sparse matrix multiplication server-side by leveraging Accumulo's native, high-performance iterators. We compare the mathematics and performance of inner and outer product implementations, and show how an outer product implementation achieves optimal performance near Accumulo's peak write rate. We offer our work as a core component to the Graphulo library that will deliver matrix math primitives for graph analytics within Accumulo.Comment: To be presented at IEEE HPEC 2015: http://www.ieee-hpec.org

On the eigenvalues of distance powers of circuits

Taking the d-th distance power of a graph, one adds edges between all pairs of vertices of that graph whose distance is at most d. It is shown that only the numbers -3, -2, -1, 0, 1, 2d can be integer eigenvalues of a circuit distance power. Moreover, their respective multiplicities are determined and explicit constructions for corresponding eigenspace bases containing only vectors with entries -1, 0, 1 are given.Comment: 14 page

Fractional Perfect b-Matching Polytopes. I: General Theory

The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed nonnegative number b_v. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.Comment: 37 page