5,493 research outputs found
Independence densities of hypergraphs
We consider the number of independent sets in hypergraphs, which allows us to
define the independence density of countable hypergraphs. Hypergraph
independence densities include a broad family of densities over graphs and
relational structures, such as -free densities of graphs for a given graph
In the case of -uniform hypergraphs, we prove that the independence
density is always rational. In the case of finite but unbounded hyperedges, we
show that the independence density can be any real number in Finally,
we extend the notion of independence density via independence polynomials
Approximating the Largest Root and Applications to Interlacing Families
We study the problem of approximating the largest root of a real-rooted
polynomial of degree using its top coefficients and give nearly
matching upper and lower bounds. We present algorithms with running time
polynomial in that use the top coefficients to approximate the maximum
root within a factor of and when and respectively. We also prove corresponding
information-theoretic lower bounds of and
, and show strong lower
bounds for noisy version of the problem in which one is given access to
approximate coefficients.
This problem has applications in the context of the method of interlacing
families of polynomials, which was used for proving the existence of Ramanujan
graphs of all degrees, the solution of the Kadison-Singer problem, and bounding
the integrality gap of the asymmetric traveling salesman problem. All of these
involve computing the maximum root of certain real-rooted polynomials for which
the top few coefficients are accessible in subexponential time. Our results
yield an algorithm with the running time of for all
of them
Graph presentations for moments of noncentral Wishart distributions and their applications
We provide formulas for the moments of the real and complex noncentral
Wishart distributions of general degrees. The obtained formulas for the real
and complex cases are described in terms of the undirected and directed graphs,
respectively. By considering degenerate cases, we give explicit formulas for
the moments of bivariate chi-square distributions and Wishart
distributions by enumerating the graphs. Noting that the Laguerre polynomials
can be considered to be moments of a noncentral chi-square distributions
formally, we demonstrate a combinatorial interpretation of the coefficients of
the Laguerre polynomials
Higher order matching polynomials and d-orthogonality
We show combinatorially that the higher-order matching polynomials of several
families of graphs are d-orthogonal polynomials. The matching polynomial of a
graph is a generating function for coverings of a graph by disjoint edges; the
higher-order matching polynomial corresponds to coverings by paths. Several
families of classical orthogonal polynomials -- the Chebyshev, Hermite, and
Laguerre polynomials -- can be interpreted as matching polynomials of paths,
cycles, complete graphs, and complete bipartite graphs. The notion of
d-orthogonality is a generalization of the usual idea of orthogonality for
polynomials and we use sign-reversing involutions to show that the higher-order
Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are
d-orthogonal. We also investigate the moments and find generating functions of
those polynomials.Comment: 21 pages, many TikZ figures; v2: minor clarifications and addition
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