(2.13) A^{2} = kI +A + (J - I - A).

Also, since **G** is regular:

(2.14) AJ = JA = kJ.

Conversely, a strongly regular graph can be defined as a graph (not complete or null) whose adjacency matrix satisfies (2.13) and (2.14).

The all 1 vector **j** is an eigenvector of both A and J with eigenvalues k and n respectively.
Applying (2.13) to this vector, we obtain

As A and J are commuting real symmetric matrices, they can be simultaneously diagonalized
by an orthogonal matrix. Thus any further eigenvectors of A are orthogonal to **j**, and so, are
eigenvectors of J with eigenvalue 0. Let **m** be such an eigenvector, with eigenvalue þ (wrt A).
Then

þ

0 = Trace(A) = k + fr + gs.

(2.16) **Theorem**. The numbers

are non-negative integers.

(2.17)**Example**: The integrality conditions applied to a (6u-3, 2u, 1,u) srg yields that

**Def**: (n,k,,) is a **Type I** parameter set if (n-1)( -) = 2k. Then since n > 1 + k and ,
we have that n = 4 + 1, k = 2 and = - 1.

(2.18) **Theorem**. If a type I srg on n vertices exists, then n is the sum of two integer squares.

**Note**: The Paley graphs are of type I.

**Def**: **Type II** parameters are those satisfying the integrality condition "normally". A Type I srg
is of Type II iff n is a square.

An example of this last possibility is given by the graph L_{2}(3) which is isomorphic to P(9).

Note that a graph Γ is regular if and only if **j** is an eigenvector of A. Furthermore, for regular graphs will be A-invariant, that is, vectors orthogonal to **j** are mapped by A to vectors orthogonal to **j**. When we restrict A to the subspace , if Γ is strongly regular then there are just two eigenvalues (r and s). If an adjacency matrix A has just two eigenvalues on , then (A - rI)(A - sI) is a multiple of J. This leads to:

(2.19) **Proposition**. The regular graph Γ with adjacency matrix A is strongly regular if and only if A restricted to has exactly two eigenvalues.

*Proof*: By the above remarks, (A - rI)(A - sI) = mJ, and so we have A^{2} = -rsI + (r+s)A + mJ = (k - m)I + (λ - m)A + mJ = kI + λA + m(J - I - A), thus satisfying (2.13).

We now consider some further necessary conditions on the parameters of an srg.

**Definition**: A square matrix A is ** positive semidefinite** if for all x ε V, x

(2.22) **Lemma**. Let A = (a_{ij}) be a positive semidefinite symmetric real n × n matrix of rank d. Then there are vectors v_{1},v_{2},...,v_{n} in ^{d} such that a_{ij} = <v_{i},v_{j}> for i,j = 1,...,n.

*Proof*: There is a basis with respect to which the quadratic form Q_{A}(x) is the sum of squares. Letting P be the change of basis matrix (real and invertible) gives,

A is called the ** Gram matrix** of the set v

(2.23) **Theorem**. Let Γ be a strongly regular graph on n vertices, having the properties that Γ and its complement are both connected and that the adjacency matrix of Γ has an eigenvalue of multiplicity f (greater than 1). Then n ½ f(f+3).

*Proof*: The adjacency matrix A has 3 distinct eigenspaces, and any matrix having these eigenspaces is a linear combination of I, A and J-I-A. In particular, there is such a linear combination E having eigenvalue 1 on the given f-dimensional eigenspace and 0 on its complement. Then E is positive semidefinite, and so is the Gram matrix of a set S of vectors of ^{f}. Since E = aI + bA + c(J-I-A), any vector in S has length a^{½}, and two vectors in S make an angle cos^{-1}(b/a) or cos^{-1}(c/a). The vectors are all distinct since neither Γ nor its complement is a disjoint union of complete graphs. We may normalize to assume that a = 1, that is, S is a subset of the unit sphere Ω.

For v ε S, let f_{v} : Ω be the function defined by

This bound is called the * absolute bound*.

**Example**: Continuing with example (2.17). If u = 11 we obtain f,g = (33 - 2) ± (33 - 10 + 1) = 31 ± 24. Taking the smaller value for f, that is f = 7, the absolute bound gives n 35, but n = 55 + 7 + 1 = 63, so this value of u is excluded.

**Definition**: The Hadamard product of two n × n matrices A and B, denoted AºB, is the entrywise product, that is, the (i,j)th entry is a_{ij}b_{ij}.

(2.25) **Lemma**. Let A and B be positive semi-definite real symmetric matrices. Then AºB is positive semi-definite.

*Proof*: The Kronecker product of A and B is positive semidefinite since its eigenvalues are all products of an eigenvalue of A with an eigenvalue of B. The Hadamard product is a principal submatrix of the Kronecker product, and so, represents the Kronecker product restricted to a subspace. Thus, AºB is positive semidefinite.

(2.26) **Theorem**. Let Γ be a strongly regular graph which is connected and whose complement is connected. If Γ has eigenvalues k,r and s, then

(a) (r+1)(k+r+2rs) (k+r)(s+1)^{2};

(b) (s+1)(k+s+2rs) (k+s)(r+1)^{2}.

*Proof*: The idempotent matrix E of the proof of (2.23) is positive semidefinite, so EºE is positive semidefinite. But

These bounds are known as the ** Krein conditions**.

(2.27) **Proposition**. (a) The graph Γ is associated with a polarity of a symmetric 2-design with no absolute points if and only if it is strongly regular with μ = λ.

(b) Γ is associated with a polarity of a symmetric 2-design with every point absolute if and only if it is strongly regular with μ = λ + 2.

*Proof*: If no point is absolute, then x^{σ} y^{σ} = Γ(x) Γ(y) has cardinality λ, whether or not x and y are adjacent. Conversely, if Γ is strongly regular with μ = λ, then (X, {Γ(x): x ε X}) is a symmetric 2-design and σ : x Γ(x) a polarity without absolute points. (b) is obtained by applying (a) to the complementary design and graph.

**Definition**: A strongly regular graph with parameters (v,k,λ,λ) is called a ** (v,k,λ) graph**.

(2.28) **Proposition**. For fixed λ, there are only finitely many (v,k,λ) graphs.

*Proof*: Let Γ be such a graph. Since μ = λ, Γ must be of type II with respect to the integrality condition. This implies that 4(k- λ) = s^{2} for some integer s, and s divides -2k. Letting t = s/2, we have k = λ + t^{2} with t dividing k. But this implies that t divides λ, so we have that t λ. Thus, k λ + λ^{2} and therefore it follows from (2.6) that v λ^{2}(λ + 2).

*Note*: The extremal case v = λ^{2}(λ+2), k = λ(λ+1) occurs for all prime power values of λ and we will see examples later. Also note that L_{2}(4) is a (16,6,2) graph. We shall see later that there is just one other (16,6,2) graph, the Shrikhande graph. It turns out that these two graphs are associated with different polarities of the same 2-(16,6,2) design.

For the strongly regular graphs associated with polarities in case (b) of (2.27), no such finiteness condition is known. In the smallest case, λ = 0 and μ = 2, there are only three known graphs:

(a) CP(2), having parameters (4,2,0,2);

(b) the Clebsch graph, with parameters (16,5,0,2); and

(c) the Gewirtz graph, with parameters (56,10,0,2).

All these graphs are uniquely determined by their parameters. They are associated with polarities of the biplanes 2-(4,3,2), 2-(16,6,2) and 2-(56,11,2). It is known that there are exactly 3 non-isomorphic biplanes with k = 6 and at least five with k = 11, but the other designs do not admit polarities with every point absolute.

**Example**: Let V be a 3-dimensional vector space and define Q(x_{1},x_{2},x_{3}) = x_{1}x_{2} + x_{3}^{2}. The polarization of Q is

This Q is non-singular, since f(x,y) = 0 for y = (0,1,0) and (1,0,0) shows that x

For a non-singular quadratic form Q, we obtain a graph Γ with vertex set {x ε V : Q(x) = 0, x not 0}, in which x and y are adjacent iff f(x,y) = 0.

**Example (cont.)**: For our example, the vertex set is a = (0,1,0), b = (1,0,0) and c = (1,1,1). Since f(a,b) = f(a,c) = f(b,c) = 1, the graph obtained is a null graph on 3 vertices.

It can be shown that, except for some small trivial examples (such as ours), Γ is strongly regular. Furthermore, Γ has the triangle property: any edge {x,y} is contained in a triangle {x,y,z} having the property that any further vertex is joined to one or all of x,y,z. (Since Q(x) = Q(y) = f(x,y) = 0, Q(x+y) = 0 and x+y is the required third vertex of the triangle. The triangle property is a consequence of the equation f(x,w) + f(y,w) + f(x+y, w) = 0.)

(2.30) **Theorem** (Shult, Seidel): A non-null graph with the triangle property, in which no vertex is joined to all others, is obtained as above from a non-singular quadratic form over GF(2).