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Abstract. We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a self-adjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a max-degree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount.

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Nanoporous bioglass containing silver (n-BGS) was fabricated using the sol-gel method, with cetyltrimethyl ammonium bromide as template. The results showed that n-BGS with nanoporous structure had a surface area of 467 m 2 /g and a pore size of around 6 nm, and exhibited a significantly higher water absorption rate compared with BGS without nanopores. The n-BGS containing small amounts of silver (Ag) had a slight effect on its surface area. The n-BGS containing 0.02 wt% Ag, without cytotoxicity, had a good antibacterial effect on Escherichia coli, and its antibacterial rate reached 99% in 12 hours. The n-BGS's clotting ability significantly decreased prothrombin time (PT) and activated partial thromboplastin time (APTT), indicating n-BGS with a higher surface area could significantly promote blood clotting (by decreasing clotting time) compared with BGS without nanopores. Effective hemostasis was achieved in skin injury models, and bleeding time was reduced. It is suggested that n-BGS could be a good dressing, with antibacterial and hemostatic properties, which might shorten wound bleeding time and control hemorrhage.