Binomial Cayley Graphs and Applications to Dynamics on Finite Spaces

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.