Using two graph invariants arising from Chung and Yau's discrete Green's function, we derive explicit formulas and new estimates of hitting times of random walks on weighted graphs through the enumeration of spanning trees.
The Chung–Yau graph invariants were originated from Chung–Yau's work on discrete Green's function. We show how they could be used to derive new explicit formulas and estimates for hitting times of random walks. We also apply them to study graphs with symmetric hitting times.
Abstract. This paper is concerned with how the QR factors change when a real matrix A suffers from a left or right multiplicative perturbation, where A is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in the QR factors in norm are no bigger than a small constant multiple of the norm of the difference between the perturbation and the identity matrix. One of common cases for a left multiplicative perturbation case naturally arises from computing the QR factorization of A. The newly established bounds can be used to explain the accuracy in the computed QR factors. For a right multiplicative perturbation, the bounds on the relative changes in the QR factors are still dependent upon the condition number of the scaled R-factor, however. Some "optimized" bounds are also obtained by taking into account certain invariant properties in the factors.
We apply Chung-Yau invariants to calculate the number of spanning trees of a complete multipartite graph. We also give explicit formulas for hitting times of random walks on a complete multipartite graph and prove that it has symmetric hitting times if and only it is vertex-transitive.
A Hausdorff topological group topology on a group G is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on G. For every compact metrizable space X containing an open n-cell, n ≥ 2, the homeomorphism group H(X) has no minimum Hausdorff group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space X containing a dense open one-manifold, H(X) has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology. * 2010 Mathematics Subject Classification: 20B27, 22A05, 22F50, 54F05, 54H15, 57S05
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