Combinatorial Identities from the Spectral Theory of Quantum Graphs

Abstract

The purpose of this paper is to present a newly discovered link between three seemingly unrelated subjects—quantum graphs, the theory of random matrix ensembles and combinatorics. We discuss the nature of this connection, and demonstrate it in a special case pertaining to simple graphs, and to the random ensemble of ${2\times 2}$ unitary matrices. The corresponding combinatorial problem results in a few identities, which, to the best of our knowledge, were not proven previously.