Graph Theory: Chemical and Biological Applications
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JENNIFER EDMOND, Syracuse University
Chain Configurations of 4-Clusters in Fullerenes  [PDF]

Fullerenes can be considered to be either a molecule of pure carbon or the trivalent plane graph with all pentagonal or hexagonal faces that models the molecule. Pairs of pentagonal faces can be connected by sequences of edges and faces in the hexagonal tessellation that we call chains. These chains are used to compute the Clar number, a parameter of much interest to chemists as it appears to indicate information about the stability of the molecule. Clusters in a fullerene are formed when a number of pentagons are isolated from the others. Pairing the four pentagons in a 4-cluster requires chains in complex configurations. The chains often wrap around one another each simultaneously affecting the other. We discuss how the configurations of 4-clusters enter into the computation of the Clar number.

ELIZABETH HARTUNG, Massachusetts College of Liberal Arts
Pairwise incompatibility of predictors of stability for graphene patches  [PDF]

Graphene patches, or benzenoids, are plane graphs with all interior faces hexagonal and with vertices of degree 2 or 3, such that the only degree-2 vertices bound the outside face. Three classic parameters thought to be related to their stability are the number of Kekule Structures (perfect matchings), the Fries number, and the Clar number. We find pairs of benzenoids X and Y that are incompatible in that one parameter is larger for benzenoid X and the other is larger for benzenoid Y. We find the smallest incompatible pairs with respect to the number of vertices, and also show that the gap between these parameters can be arbitrarily large. Co-authors: James Chapman, Judith Foos, Andrew Nelson, Aaron Williams

Network Alignment  [PDF]

Network is a powerful way of representing and analyzing biological data. One important question is, how much two different biological networks of same category are similar to each other. Mathematical translation of this question is network alignment.

In general, network alignment identifies a bijection between the full (or partial) vertex sets of two networks such that the size of corresponding common subgraph is maximized. This problem is closely related to the quadratic assignment problem and is known to be NP-hard not only to solve, but also to approximate. Because of its applications in different areas like systems biology or social sciences, finding efficient algorithm to approximate the optimal alignment is essential problem. In this talk I will first mention applications of this problem in biology then review some known algorithms which are mainly based on spectral techniques and state some of our new results at the end.

PRATIBHA, Indian Institute of Technology Roorkee, India
Two dimensional model of pulsatile flow of dusty fluid in pulmonary region  [PDF]

Pratibha, Jyoti Kori,

IITRoorkee, India, Email: pratifma@iitr.ac.in

We present a mathematical model of non-spherical nano particulate suspension and deposition underneath periodic breathing in pulmonary region for different stages of lungs. The pulsatile flow behavior inside airways with sinusoidal wall oscillation through a non-Darcian porous medium is studied. Possible effects of non-spherical nan particulate is modeled through aerodynamic diameter concept and considering the drag force term in the translational momentum equation. The transport equations are formulated in a two dimensional coordinate system using boundary layer theory and solved numerically. General solution of governed unsteady non-linear Navier-Stokes equations is obtained for inlet Reynolds number $0.01 \le Re \le 1.2$, Womersley number ($\alpha$) is of $O(4)$ or less, Forchsheimer number and particle shape factor $\le 1000$. Results for velocity of fluid and dust particles and wall shear stress distribution are discussed to understand the critical condition of interstitial lung diseases.