Harris Kwong

Some new characterizations of Hamiltonian cycles in triangular grid graphs

Abstract In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate the thermodynamics of protein folding. In this paper, we present new characterizations of the Hamiltonian cycles in labeled triangular grid graphs, which are graphs constructed from rectangular grids by adding a diagonal to each cell. By using these characterizations and implementing the computational method outlined here, we confirm the existing data, and obtain some new results that have not been published. A new interpretation of Catalan numbers is also included.

Two determinants with Fibonacci and Lucas entries

In this short note, we study two families of determinants the entries of which are linear functions of Fibonacci or Lucas numbers. The results are rather simple, and the two determinants only differ by a constant.

A limit conjecture on the number of Hamiltonian cycles on thin triangular grid cylinder graphs

Abstract We continue our research in the enumeration of Hamiltonian cycles (HCs) on thin cylinder grid graphs Cm × Pn+1 by studying a triangular variant of the problem. There are two types of HCs, distinguished by whether they wrap around the cylinder. Using two characterizations of these HCs, we prove that, for fixed m, the number of HCs of both types satisfy some linear recurrence relations. For small m, computational results reveal that the two numbers are asymptotically the same. We conjecture that this is true for all m ≥ 2.

Evaluating Simultaneous Integrals.

Abstract Many integrals require two successive applications of integration by parts. During the process, another integral of similar type is often invoked. We propose a method which can integrate these two integrals simultaneously. All we need is to solve a linear system of equations.