An *m* by *n* **rook graph** is formed by associating a node with each square of an *m* by *n* chess board and connecting two nodes with an edge if a rook can legally move from one to the other. That is, two nodes are connected if they are either in the same row or in the same column.

A **Paley graph** of order *q* is a graph with *q* nodes where *q* is a prime power and congruent to 1 mod 4. Label each node with an element of the finite field *F* with *q* elements and connect two nodes with an edge if the difference between their labels is a quadratic residue. That is, for nodes labeled *a* and *b*, connect these nodes if there exist an *x* in *F* such that *a* – *b* = *x*².

We’ll look at a graph which is both a rook graph and a Paley graph: the 3 by 3 rook graph is the same as the Paley graph of order 9.

## Rook graph 3 x 3

Constructing the rook graph is easy. If we number our 3 by 3 chess board as follows

1 2 3 4 5 6 7 8 9

then we connect 1 to 2, 3, 4, and 7. We connect 2 to 1, 3, 5, and 8. Etc. This gives us the following graph.

(The graph was made with Sketchviz introduced here.)

## Paley graph of order 9

They Paley graph is a little more complicated to construct. First we need to explain what a field with 9 elements is.

A field with 3 elements is just integers mod 3. We can think of the field with 9 elements as the analog of complex numbers: numbers of the form *a* + *bi* where *a* and *b* are integers mod 3. As with the complex numbers, *i*² = -1.

When we square each of the nine elements of our field, we get four distinct non-zero results: 1, 2, *i*, and 2*i*. For example, *i* is one of our squares because

(1 + 2*i*)(1 + 2*i*) = (1 – 4) + 4*i* = *i*

Here we used -3 = 0 mod 3 and 4 = 1 mod 3.

Since our squares are 1, 2, *i*, and 2*i*, that says 0 is connected to these four values. Then we connect 1 to 2, 0, 1 + *i* and 1 + 2*i* because each of these numbers minus 1 is a square. If we keep doing this for all nine nodes, we get the following graph.

## Isomorphism

It’s clear that the two graphs above are relabelings of each other. We can see this explicitly by relabeling our chess board as follows:

0 i 2i 1 1+i 1+2i 2 2+i 2+2i

Moving horizontally adds *i*, which is a square, and moving vertically adds 1, which is a square. So the Paley connections are rook connections.

## Uniqueness

Are there more rook graphs that are isomorphic to Paley graphs? No, this is the only one.

Every pair of vertices in a rook graph that are not neighbors have two common. If the points with coordinates (*a*, *b*) and (*c*, *d*) are on different rows and columns, then both are connected to (*a*, *d*) and (*c*, *b*).

In a Paley graph of order *q*, every pair of nodes that aren’t neighbors have (*q* – 1)/4 common neighbors. For a Paley graph to be isomorphic to a rook graph, we must have (*q* – 1)/4 = 2, and so *q* = 9.

Thanks. I’m fixing it now.

Technically there’s one other case: the 1×2 rook graph is isomorphic to the Paley graph of order 2 (both consist of 2 nodes joined by an edge). This doesn’t contradict your counting argument because there are no pairs of non-adjacent nodes.

The rook graph shows 5 edges for node 8 and node 4, because node 8 and node 4 are show connected. I didn’t expect node 8 and 4 to be connected?