871 research outputs found
Tree decomposition by eigenvectors
AbstractIn this work a composition–decomposition technique is presented that correlates tree eigenvectors with certain eigenvectors of an associated so-called skeleton forest. In particular, the matching properties of a skeleton determine the multiplicity of the corresponding tree eigenvalue. As an application a characterization of trees that admit eigenspace bases with entries only from the set {0,1,−1} is presented. Moreover, a result due to Nylen concerned with partitioning eigenvectors of tree pattern matrices is generalized
Prime power divisors of binomial coefficients
[No abstract available
Extremal energies of integral circulant graphs via multiplicativity
AbstractThe energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. Integral circulant graphs can be characterised by their order n and a set D of positive divisors of n in such a way that they have vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd(a-b,n)∈D}. Among integral circulant graphs of fixed prime power order ps, those having minimal energy Eminps or maximal energy Emaxps, respectively, are known. We study the energy of integral circulant graphs of arbitrary order n with so-called multiplicative divisor sets. This leads to good bounds for Eminn and Emaxn as well as conjectures concerning the true value of Eminn
Integral circulant graphs of prime power order with maximal energy
The energy of a graph is the sum of the moduli of the eigenvalues of its
adjacency matrix. We study the energy of integral circulant graphs, also called
gcd graphs, which can be characterized by their vertex count n and a set D of
divisors of n in such a way that they have vertex set Zn and edge set {{a, b} :
a, b in Zn; gcd(a - b, n) in D}. Using tools from convex optimization, we study
the maximal energy among all integral circulant graphs of prime power order ps
and varying divisor sets D. Our main result states that this maximal energy
approximately lies between s(p - 1)p^(s-1) and twice this value. We construct
suitable divisor sets for which the energy lies in this interval. We also
characterize hyperenergetic integral circulant graphs of prime power order and
exhibit an interesting topological property of their divisor sets.Comment: 25 page
The maximal energy of classes of integral circulant graphs
The energy of a graph is the sum of the moduli of the eigenvalues of its
adjacency matrix. We study the energy of integral circulant graphs, also called
gcd graphs, which can be characterized by their vertex count and a set
of divisors of in such a way that they have vertex set
and edge set . For a fixed prime power and a fixed divisor set size , we analyze the maximal energy among all matching integral circulant
graphs. Let be the elements of .
It turns out that the differences between the exponents of
an energy maximal divisor set must satisfy certain balance conditions: (i)
either all equal , or at most the two differences
and may occur; %(for a certain depending on and ) (ii)
there are rules governing the sequence of consecutive
differences. For particular choices of and these conditions already
guarantee maximal energy and its value can be computed explicitly.Comment: Discrete Applied Mathematics (2012
On primes not dividing binomial coefficients
We prove that [Formula Omitted] thus dealing with open problems concerning divisors of binomial coefficients. © 1993, Cambridge Philosophical Society. All rights reserved
Irrationality Criteria for Mahler′s Numbers
AbstractFor positive integers m and h ≥ 2, let (m)h denote the finite sequence of digits of m written in h-ary notation. It is known that the real number ah(g) = 0 · (gn1)h(gn2)h(gn3)h... with g ≥ 2, h ≥ 2 is irrational, if the sequence (ni) of non-negative integers is unbounded. We study the case where (ni) is bounded, and prove several irrationality criteria
Prime power divisors of binomial coefficients: Reprise
[No abstract available
An asymptotic formula for a-th powers dividing binomial coefficients
[No abstract available
A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations
We introduce a space–time discontinuous Galerkin (DG) finite element method for the incompressible Navier–Stokes equations. Our formulation can be made arbitrarily high order accurate in both space and time and can be directly applied to deforming domains. Different stabilizing approaches are discussed which ensure stability of the method. A numerical study is performed to compare the effect of the stabilizing approaches, to show the method’s robustness on deforming domains and to investigate the behavior of the convergence rates of the solution. Recently we introduced a space–time hybridizable DG (HDG) method for incompressible flows [S. Rhebergen, B. Cockburn, A space–time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys. 231 (2012) 4185–4204]. We will compare numerical results of the space–time DG and space–time HDG methods. This constitutes the first comparison between DG and HDG methods
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