Students (Win2022): Lily Gibbs, Griffin Golias

Mentors: Peter Gylys-Colwell, Junaid Hasan, Harry Richman

- The problem of counting spanning trees on a graph is of interest in combinatorics, probability, and statistical physics. On a regular 2-dimensional lattice, the number of spanning trees is known to grow exponentially. The exponential rate is known for the square and triangular lattices, but not for other lattices. The goal of this project is to calculate the asymptotic growth of the number of spanning trees on the Kagome lattice. Students will learn about graph Laplacians, Kirchhoff's matrix-tree theorem, asymptotic analysis, and random walks on graphs. Computational tools may also be used.

- Kagome (or trihexagonal) lattice Wikipedia
- Intro to Algebraic Graph Theory, C. O. Aguilar
- Modern Graph Theory, B. Bollobas
- Combinatorics of Electrical Networks, D. G. Wagner
- On the number of spanning trees on various lattices, E. Teufl and S. Wagner
- Graph polynomials and their applications I. The Tutte polynomial, J. Ellis-Monaghan and C. Merino
- The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, A. Sokal
- Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs, M. Abért and T. Hubai