Binomial Cayley Graphs and Applications to Dynamics on Finite Spaces

TL;DR

This paper introduces binomial Cayley graphs derived from classical Cayley graphs, analyzes their spectral and combinatorial properties, and applies these findings to understand dynamics in finite particle systems.

Contribution

It defines binomial Cayley graphs, explores their properties for symmetric and cyclic groups, and links spectral features to permutation patterns and finite dynamical systems.

Findings

Spectral analysis reveals eigenvalue multiplicities relate to permutation structures.

Established connections between null eigenvalues and longest increasing subsequences.

Applied graph properties to describe degeneracy in finite particle dynamics.

Abstract

Binomial Cayley graphs are obtained by considering the binomial coefficient of the weight function of a given Cayley graph and a natural number. We introduce these objects and study two families: one associated with symmetric groups and the other with powers of cyclic groups. We determine various combinatorial properties of these graphs through the spectral analysis of their adjacency matrices. In the case of symmetric groups, we establish a relation between the multiplicity of the null eigenvalue and longest increasing sub-sequences of permutations by means of the RSK correspondence. Finally, we consider dynamical arrangements of finitely many elements in finite spaces, which we refer to as particle-box systems. We apply the results obtained on binomial Cayley graphs in order to describe their degeneracy.