Understanding biomolecules---their structures, dynamics, and
interactions with solvent---is essential to revealing mechanisms
and functions of biological systems. While atomistic simulations
that treat both solvent and solute molecules explicitly are usually
more accurate, implicit or continuum solvent models for biomolecules
are far more efficient. With an implicit solvent, the free energy
and structure of an underlying solvation system is described
through the solute particles and the interface that separates the
solutes and solvent.

Dzubiella, Swanson and McCammon [Phys. Rev. Lett.104, 527 (2006)
and J. Chem. Phys. 124, 084905 (2006)] developed a class of
variational implicit-solvent models. Central in these models is a
free-energy functional of all admissible solute-solvent interfaces,
coupling both nonpolar and polar contributions of an underlying
system. An energy-minimizing interface then defines an equilibrium
solute-solvent interface. Cheng et al. [J. Chem. Phys. 127, 084503
(2007)] developed a robust level-set method for numerically
capturing such interfaces.

In this talk, I will give an overview of the recent development
of variational implicit-solvent approach for solvation systems.
I will point out how various kinds of mathematical concepts and
techniques from differential geometry and partial differential
equations can be applied to this approach.

Joint work with Jianwei Che, Li-Tien Cheng, Joachim Dzubiella,
J. Andy McCammon, and Yang Xie.

We will introduce regular graphs, and explore the eigenvalues associated to these graphs. We will then see what the eigenvalues of a graph tell us about its expansion. And, why this is actually interesting. Finally we will discuss irregular graphs and what statements we may make about their eigenvalues.