TITLE: Algebraic properties of graph of chess pieces
QUESTION [11 upvotes]: 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?

REPLY [2 votes]: Wikipedia says the chromatic polynomial is "studied in algebraic graph theory".  So here's an answer involving the chromatic polynomial of the rook's graph.
The rook's graph is $K_8 \Box K_8$, i.e., the Cartesian product of $K_8$ and $K_8$.
The chromatic polynomial $P(K_8 \Box K_8;n)$ of the rook's graph is the number of $8 \times 8$ generalized Latin squares, where we can use $n$ symbols (without repeated symbols in any row or column).  In particular, evaluating it at its chromatic number...
$$P(K_8 \Box K_8;8) = \text{number of Latin squares of order $8$}.$$