247 research outputs found
A formula for the number of spanning trees in circulant graphs with non-fixed generators and discrete tori
We consider the number of spanning trees in circulant graphs of
vertices with generators depending linearly on . The matrix tree theorem
gives a closed formula of factors, while we derive a formula of
factors. Using the same trick, we also derive a formula for the
number of spanning trees in discrete tori. Moreover, the spanning tree entropy
of circulant graphs with fixed and non-fixed generators is compared.Comment: 8 pages, 2 figure
Which Digraphs with Ring Structure are Essentially Cyclic?
We say that a digraph is essentially cyclic if its Laplacian spectrum is not
completely real. The essential cyclicity implies the presence of directed
cycles, but not vice versa. The problem of characterizing essential cyclicity
in terms of graph topology is difficult and yet unsolved. Its solution is
important for some applications of graph theory, including that in
decentralized control. In the present paper, this problem is solved with
respect to the class of digraphs with ring structure, which models some typical
communication networks. It is shown that the digraphs in this class are
essentially cyclic, except for certain specified digraphs. The main technical
tool we employ is the Chebyshev polynomials of the second kind. A by-product of
this study is a theorem on the zeros of polynomials that differ by one from the
products of Chebyshev polynomials of the second kind. We also consider the
problem of essential cyclicity for weighted digraphs and enumerate the spanning
trees in some digraphs with ring structure.Comment: 19 pages, 8 figures, Advances in Applied Mathematics: accepted for
publication (2010) http://dx.doi.org/10.1016/j.aam.2010.01.00
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