Strongly Regular Graphs

Our graphs will be simple undirected graphs (no loops or multiple edges). Graphs do not make interesting designs. A general graph is a 0-design with k = 2. A regular graph is a 1- design and the only 2-designs come from complete graphs. The interesting connections lie in other directions.

We start with an example:

(2.3) Theorem. Let G be a graph with the following property:
(*) Any edge {x,y} is contained in a triangle (3-clique) {x,y,z} having the property that any other vertex is adjacent to exactly one of x,y or z.

Then G is isomorphic to one of the following:

1. a null graph;
2. a "windmill" (a number of triangles with a common vertex);
3. three special regular graphs having 9, 15 and 27 vertices respectively.

Let x be any vertex of G. The closed neighborhood of x, {x}G(x), is a windmill, so x has even degree, 2u say, with u 2 (if u =1 then the graph is a connected graph with all vertices having degree 2, i.e., a cycle, and the only cycle with triangles is the 3-cycle which is a special case of the windmill). Number the triangles containing x as T₁, T₂, ... , T_u and let T_i = {x, y_i,0, y_i,1}.

Now let z be any vertex not equal or adjacent to x. By (*), z is joined to just one of y_i,0 and y_i,1, say y_i,ê(i), for i = 1,...,u. We can use the function f_z : {1, ..., u}{0,1} defined by

Let (x) be the set of non-neighbors of x. We next show:

1. If z, z' (x) are adjacent, then f_z and f_z' agree in exactly one position.
2. If z, z'(x) are nonadjacent but have a common neighbor in (x), then the functions agree in all but two positions.
3. If z, z'(x) are non-adjacent and have no common neighbor in (x), then the functions agree.

Pf of (1): Here z and z' are in a triangle whose third vertex is a neighbor of x, say y_i,ê(i), so f_z(i) = f_z'(i) = ê(i). But z and z' can have no other common neighbors, so the functions can not agree in any other position. (A second common neighbor would contradict (*)).

Note that for any z(x) and any i, there is a unique neighbor z' of z for which f_z(i) = f_z'(i).

Pf of (2): If z" is a common neighbor, the result follows by applying (1) to the pairs {z, z"} and {z",z'}.

Pf of (3): If ê = f_z(i)f_z'(i), then z' is joined to a vertex of the unique triangle containing z and y_i,ê, at the third point say z". But then z" is a common neighbor in(x).

Now suppose that u 3. Take z(x) and let (z,z',z",z"') be a path of length 3 in(x) such that f_z(i) = f_z'(i), f_z'(j) = f_z''(j) and f_z''(k) = f_z'''(k), where i, j, k are three distinct coordinates. Then f_z and f_z''' agree in just these coordinates. The only way this can happen under (1)-(3) is that z and z"' are non-adjacent and 3 is either 2 less or equal to the number of coordinates. That is, u = 3 or u - 2 = 3. So, we have then that u = 2, 3 or 5.

In each of these cases there is enough information to construct the graph.

For u = 2, we obtain the above graph on 9 vertices with common valency 4. It can also be described as the graph obtained with vertices the 9 positions in a 3 by 3 array and adjacency defined by two positions being in the same row or column.

For u = 3, we obtain a graph which can be described as follows. The vertices are all the 2- sets of a 6-set (i.e. the edges of a K₆) and two vertices are adjacent if the 2-sets are disjoint. Alternatively, we are describing the complement of the edge graph of K₆. There are 15 vertices and the graph is regular of degree 6.

For u = 5, we obtain the Schläfli graph, whose description is best given by the functions of the theorem. Each function which has an even number of ones occurs exactly once as a label and no others (there are 1 + 10 + 5 = 16 of these). x and its neighborhood give 1 + 10 more vertices for a total of 27. The graph is regular of degree 10.

These are special cases of strongly regular graphs which we now define.

(2.4) Definition. A strongly regular graph with parameters (n, k,,) is a graph G with n vertices, not complete or null, in which the number of common neighbors of x and y is k,, or according as x and y are equal, adjacent or non-adjacent respectively. Note that the number does not depend on x and y - only their relationship to one another; and the equality statement is just a fancy way of including the fact that the graph is regular of degree k.

We can restate (2.3) as follows:

(2.5) Proposition. Let G be a strongly regular graph with parameters (6u-3,2u,1,u). Then u = 2,3 or 5 and there is a unique graph for each value of u.

(2.6) Proposition. If a strongly regular graph has parameters (n,k,,), then

k(k - - 1) = (n - k - 1).

Pf: Consider the edges {y,z} with yG(x) and zG(x). As the degree of x is k, there are k choices for y. Since the number of common neighbors of x and y is, there are (k - (+ 1)) neighbors of y which are not adjacent to x. This count gives the LHS of the equation. Since there are n vertices of which 1 + k are x or its neighbors, there are n - k - 1 choices for z. Since z and x are non-adjacent, there are choices for y.

(2.7) Proposition. The complement of a strongly regular graph is strongly regular.

Pf: Clearly the complement is a regular graph of valency n - k - 1. The constancy of the other parameters follows easily and the values can be calculated from Inclusion-Exclusion. In the complement the number of common neighbors of adjacent vertices is (n-2) - 2k + = n - 2k + - 2, and for non-adjacent vertices n - 2k +.

Remark: The non-negativity of these parameters gives necessary conditions on the parameters of a strongly regular graph, namely n 2k - + 2 and n 2k -.

Examples:

n = ½m(m-1), k = 2(m-2), = m - 2, = 4.

The square lattice graph L₂(m) has vertex set S × S, where S is an m-set and two vertices are adjacent if they agree in one coordinate. L₂(m) is strongly regular with parameters,

n = m², k = 2(m-1), = m - 2, = 2.

The disjoint union of r complete graphs on m vertices denoted r.K_m (r,m> 1) is strongly regular with parameters,

n = rm, k = m-1, = m - 2, = 0.

The Paley graph P(q), with q a prime power 1 mod 4 has vertex set the finite field GF(q) and two vertices are adjacent if their difference is a non-zero square (-1 is a square in these fields so the relation is symmetric). P(q) is strongly regular with parameters,

n = q, k = ½(q-1) , = ¼(q - 5), = ¼(q-1).

The Clebsch graph has as vertices all subsets of {1,2,3,4,5} of even cardinality; two vertices are adjacent if their symmetric difference has cardinality 4. It is strongly regular with parameters (16,5,0,2). The subgraph on the set of non-neighbors of a vertex is the Petersen graph. Also, the subgraph of the Schläfli graph on the set of non-neighbors of a vertex is the Clebsch graph.

The Gewirtz graph is a strongly regular graph with parameters (56,10,0,2). The vertex set is a set of hyperovals in PG(2,4). There are 168 hyperovals in this projective plane and they can be partitioned into three classes of 56 hyperovals apiece with the property that two hyperovals belong to the same class iff they intersect in an even number of points. To form the Gewirtz graph, take one of the three classes as the vertices and two vertices are adjacent if the hyperovals are disjoint. For another construction of this graph in group theoretic terms the vertex set is {}PQ, where P is the set of Sylow 3-subgroups of the alternating group A₆, and Q is the set of involutions in A₆. Join to all the vertices in P; join pP to qQ whenever q^-1pq = p; join r,sQ whenever rs has order 4.

Rank 3-graphs are strongly regular.

Let G be a group of permutations of the set X. Assume that G is transitive, that is, any point of X can be mapped to any other point by some element in G. G also acts on the set X × X by (x,y) (x^g, y^g) for each g G where x^g is the image of x under the permutation g. The number of orbits of this action is called the rank of G.

Example: Consider the dihedral group of symmetries of a square <(1234),(12)(34)>. This is a transitive group on the set X = {1,2,3,4}. It's 3 orbits on X × X are: [(1,1), (2,2), (3,3), (4,4)], [(1,2), (2,3), (3,4), (4,1), (2,1), (3,2), (4,3), (1,4)], and [(1,3), (2,4), (3,1), (4,2)]. Thus, this is a rank 3 group.

The diagonal {(x,x): x in X} is always an orbit due to the transitivity of G. A rank 2 group is doubly transitive, that is, any ordered pair of distinct elements can be mapped to any other ordered pair of distinct elements. This is no longer true for higher rank groups.

Suppose that G is a rank 3 group of even order (such as the example above). Since G has even order, it will contain an involution (an element of order 2, for instance the permutation (12)(34) in our example). Some pair of distinct elements, say x and y, of X are interchanged by this involution. Let O be the orbit containing (x,y). Define a graph Γ whose vertices are the elements of X, and whose edges are the unordered pairs {z,w} where (z,w) is in O. Because O is an orbit, all the edges of Γ are equivalent, so the number of vertices adjacent to both ends of an edge must be the same for all edges. Since the non-adjacent vertex pairs are all in the other (non-diagonal) orbit, the number of vertices adjacent to both must again be a constant. The transitivity of G implies that Γ is a regular graph. Thus, Γ is a strongly regular graph. Such a graph is called a rank 3 graph. All the examples of strongly regular graphs that we have seen so far are actually rank 3 graphs, but as we shall see most strongly regular graphs are not rank 3 graphs.

The rank 3 graph corresponding to our example is the cycle C₄, which is the unique (4,2,0,2) srg.