For the purpose of this question, a *chess piece* is the King, Queen, Rook, Bishop or Knight of the game of chess. To a chess piece is attached a graph which represents the legal moves it can make on an empty chess board. Except for the bishop, the graph attached to a chess piece has 64 vertices, while the bishop's graph has 32 vertices.

These graphs satisfy a number of interesting mathematical properties from the point of view of graph theory.

- Only the rook's graph is regular, and the rook's graph is even strongly regular (a fact well-known to chess player with no knowledge of mathematics).
- The knight's graph is bipartite.
- The king's, rook's, queen's and bishop's graph are hamiltonian (and after Euler, it is very well-known though to me non-obvious
*a priori*that the knight's graphs is hamiltonian).

Less obvious properties have of course been devised by graph theorists.

Recently, I asked myself (with no other purpose than leisure) if these graphs were known to satisfy interesting algebraic properties.

A quick computation based on its strong regularity or on the observation that it is the cartesian product of two copies of the complete graph $K_8$ show that spectrum of the rook's graph is $\{14^{(1)},6^{(14)},-2^{(49)}\}$ (in particular, it is an integral graph).

Are there any other remarkable algebraic properties of the graphs of chess pieces, either of their spectra or of their group of automorphisms?