Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs

TL;DR

This paper introduces a cyclic decomposition for k-permutations to analyze the eigenvalues of arrangement graphs, providing a detailed spectral characterization and connecting it to Johnson graphs.

Contribution

It presents a novel cyclic decomposition method that yields a fine equitable partition of arrangement graphs, enabling eigenvalue computation and establishing links to Johnson graph spectra.

Findings

Eigenvalues of A(n,k) computed for small k

Any eigenvalue of J(n,k) is also an eigenvalue of A(n,k)

The smallest eigenvalue of A(n,k) is -k with high multiplicity

Abstract

The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k-1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n,k). This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of A(n,k). Consequently, we determine the eigenvalues of A(n,k) for small values of k. Finally, we show that any eigenvalue of the Johnson graph J(n,k) is an eigenvalue of A(n,k) and that -k is the smallest eigenvalue of A(n,k) with multiplicity O(n^k) for fixed k.