290 research outputs found
On the class of graphs with strong mixing properties
We study three mixing properties of a graph: large algebraic connectivity,
large Cheeger constant (isoperimetric number) and large spectral gap from 1 for
the second largest eigenvalue of the transition probability matrix of the
random walk on the graph. We prove equivalence of this properties (in some
sense). We give estimates for the probability for a random graph to satisfy
these properties. In addition, we present asymptotic formulas for the numbers
of Eulerian orientations and Eulerian circuits in an undirected simple graph
Asymptotic enumeration of Eulerian circuits for graphs with strong mixing properties
We prove an asymptotic formula for the number of Eulerian circuits for graphs
with strong mixing properties and with vertices having even degrees. The exact
value is determined up to the multiplicative error ,
where is the number of vertice
Asymptotic behavior of the number of Eulerian orientations of graphs
We consider the class of simple graphs with large algebraic connectivity (the
second-smallest eigenvalue of the Laplacian matrix). For this class of graphs
we determine the asymptotic behavior of the number of Eulerian orientations. In
addition, we establish some new properties of the Laplacian matrix, as well as
an estimate of a conditionality of matrices with the asymptotic diagonal
predominanceComment: arXiv admin note: text overlap with arXiv:1104.304
Rates of DNA Sequence Profiles for Practical Values of Read Lengths
A recent study by one of the authors has demonstrated the importance of
profile vectors in DNA-based data storage. We provide exact values and lower
bounds on the number of profile vectors for finite values of alphabet size ,
read length , and word length .Consequently, we demonstrate that for
and , the number of profile vectors is at least
with very close to one.In addition to enumeration
results, we provide a set of efficient encoding and decoding algorithms for
each of two particular families of profile vectors
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