Dusan Ziga, with support from the University by way of a Summer Research Internship, and James Bradshaw, supported by an Undergraduate Research Bursary from the London Mathematical Society, have been working on a summer research project together with Dr Philipp Lampe. Their investigation has been into what are called snake graphs (an example is shown above). Snake graphs arise in algebra and mathematical physics.
In this project James, Dusan, and Philipp wish to understand the combinatorics of snake graphs by means of computer programs and linear algebra. Based on numerical experimentations, they extended Kasteleyn’s ideas about eigenvectors of weighted adjacency matrices of horizontal snake graphs to other classes of snake graphs. The knowledge of the eigenvectors and eigenvalues makes it possible to determine the number of perfect matchings of the graph as a product of the eigenvalues.
In mathematics young talents show up relatively early and there is a long tradition of building bridges between school/university mathematics and research mathematics. Such experiences can have a very positive influence on the career of students. The summer project is modelled on the way a research-active mathematician works, with one aim being to introduce students to scientific working methodology.
James summed up his experience as follows: “I have really enjoyed the work and I have learnt a lot, and this has helped me to think about what I might want to do in the future.”
Image source: Triangular snake graph with 7 vertices from Pratik, S. and Parmar, D. (2019) Integer Cordial Labeling of Triangular Snake Graph, ResearchGate 3118-3126.