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On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras
Any finite dimensional semisimple algebra A over a field K is isomorphic to a
direct sum of finite dimensional full matrix rings over suitable division
rings. In this paper we will consider the special case where all division rings
are exactly the field K. All such finite dimensional semisimple algebras arise
as a finite dimensional Leavitt path algebra. For this specific finite
dimensional semisimple algebra A over a field K, we define a uniquely detemined
specific graph - which we name as a truncated tree associated with A - whose
Leavitt path algebra is isomorphic to A. We define an algebraic invariant
{\kappa}(A) for A and count the number of isomorphism classes of Leavitt path
algebras with {\kappa}(A)=n. Moreover, we find the maximum and the minimum
K-dimensions of the Leavitt path algebras of possible trees with a given number
of vertices and determine the number of distinct Leavitt path algebras of a
line graph with a given number of vertices.Comment: 10 pages, 5 figure
Path Laplacian operators and superdiffusive processes on graphs. I. One-dimensional case
We consider a generalization of the diffusion equation on graphs. This
generalized diffusion equation gives rise to both normal and superdiffusive
processes on infinite one-dimensional graphs. The generalization is based on
the -path Laplacian operators , which account for the hop of a
diffusive particle to non-nearest neighbours in a graph. We first prove that
the -path Laplacian operators are self-adjoint. Then, we study the
transformed -path Laplacian operators using Laplace, factorial and Mellin
transforms. We prove that the generalized diffusion equation using the Laplace-
and factorial-transformed operators always produce normal diffusive processes
independently of the parameters of the transforms. More importantly, the
generalized diffusion equation using the Mellin-transformed -path Laplacians
produces superdiffusive processes when
Directed path graphs
The concept of a line digraph is generalized to that of a directed path graph. The directed path graph of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. Several properties of are studied, in particular with respect to isomorphism and traversability
Heat capacity and phonon mean free path of wurtzite GaN
We report on lattice specific heat of bulk hexagonal GaN measured by the heat
flow method in the temperature range 20-300 K and by the adiabatic method in
the range 5-70 K. We fit the experimental data using two temperatures model.
The best fit with the accuracy of 3 % was obtained for the temperature
independent Debye's temperature {\rm K} and Einstein's
temperature {\rm K}. We relate these temperatures to the
function of density of states. Using our results for heat conduction
coefficient, we established in temperature range 10-100 K the explicit
dependence of the phonon mean free path on temperature . Above 100 K, there is the evidence of contribution of the Umklapp
processes which limit phonon free path at high temepratures. For phonons with
energy {\rm K} the mean free path is of the order 100
{\rm nm}Comment: 5 pages, 4 figure
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