308 research outputs found
Theory of Computation of Multidimensional Entropy with an Application to the Monomer-Dimer Problem
We outline the most recent theory for the computation of the exponential
growth rate of the number of configurations on a multi-dimensional grid. As an
application we compute the monomer-dimer constant for the 2-dimensional grid to
8 decimal digits, agreeing with the heuristic computations of Baxter, and for
the 3-dimensional grid with an error smaller than 1.35%.Comment: 35 pages, one pstricks and two eps figures, submitte
Generalized Sharp Bounds on the Spectral Radius of Digraphs
The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the
eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are
often tighter and are applicable to a larger class of digraphs than previously
reported bounds. Calculating the final bound pair is particularly suited to
sparse digraphs.
For strongly connected digraphs, we derive equality conditions for the
bounds, relating to the outdegree regularity of the digraph. We also prove that
the bounds hold with equality only if {\rho}(G) is the r-th root of an integer,
where r divides the index of imprimitivity of G.Comment: 12 pages and 1 figure. Nov. 30, 2012 revisio
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