k(k − a − 1) = (n − k − 1)c.
Proof: For a vertex x, consider the edges {y, z} with y adjacent to x, but z not.
There are k vertices adjacent to x, and each of them has k neighbors. One of these is x, and there are a who are also neighbor of x, so there are k − a − 1 neighbors not adjacent to x.
On the other hand, there are n − k − 1 vertices z not adjacent to x and different from x. For each such z there are c vertices y adjacent to x and z. ◻
Example: For m ≥ 4, the Johnson graph J(m, 2, 1) is strongly regular with param-eters n =m(m−1)2 , k = 2(m − 2), a = m − 2, c = 4.
Example: For a prime power q with q ≡ 1 (mod 4) (ensuring that −1 is a square in Fq), the Paley graph P(q) has vertex set Fqwith two vertices adjacent if their difference is a non-zero square in Fq.
It is strongly regular with parameters n − q, k = q−12 , a = q−54 and c = q−14 . Example: The Clebsch graph has as vertices the 16 subsets of {1, 2, 3, 4, 5} of even size. To vertices are adjacent if their symmetric difference is of size 4. It is strongly regular with parameters (16, 5, 0, 2).
Another example class of strongly regular graphs is given by orthogonal latin squares:
Let {Ai, . . . , Ak} a set of MOLS of order n. We define a graph Γ = (X, E) with X = {(i, j) ∣ 1 ≤ i, j ≤ n} and an edge given between (i, j) and (a, b) if one of the following conditions holds (compare Section VII.8):
• i = a
• j = b
• The (i, j) entry of Axequals the (a, b) entry of Ay(we allow x = y).
Theorem X.8: The graph defined by k MOLS of order n is strongly regular with parameters
(n2, (n − 1)(k + 2), n − 2 + k(k + 1), (k + 1)(k + 2)).
The proof is exercise ??.
Note X.9: IT is possible to generalize the concept of strongly regular to that of a distance regular graph in which the number of common neighbors of x, y depends on the distance of x and y.
X.2 Eigenvalues
A principal tool for studying strongly regular graphs (and generalizations) is their adjacency matrix A with Ai , j= 1 iff vertex i is adjacent to vertex j and 0 otherwise.
Note X.10: A is a symmetric matrix, thus (by the spectral theorem) it has real eigenvalues and can be diagonalized by orthogonal transformations, respectively its eigenspaces for different eigenvalues are mutually orthogonal.
Lemma X.11: If Γ is regular of degree k, then A has largest eigenvalue k with the all-one vector as associated eigenvector. k is the largest eigenvalue.
If Γ is connected, the multiplicity of k is one.
Proof: If Γ is regular, then the sum over every row of A is k, showing that the all-one vector is eigenvector for eigenvalue k.
Now let v be an eigenvector of A for eigenvalue λ ≥ k. WLOG we assume that v is scaled such that its largest entry is 1. Let i be an index of this largest entry in v.
Then the i-th entry e of Av (which by assumption must equal λ ≥ k) is the sum of the k entries of v, at indices of the vertices adjacent to vertex i. All of these entries are ≤ 1, thus e ≤ k with equality only if all these entries are 1.
This shows that k is the largest eigenvalue. Furthermore, for v to be an eigen-vector, its entries at all indices neighboring i must be 1. By induction this shows that the entries of v at indices of distance j ≥ 1 from i must be 1, showing that for a
connected Γ the multiplicity of k is one. ◻
Lemma X.12: Let Γ be a graph with adjacency matrix A. Then the i, j entry of Akis the number of walks (like path but we permit to walk back on the same route) of length k from i to j.
Proof: By induction over k it is sufficient to consider a product AB with B = Ak−1. The i, j entry of this product is ∑sai ,sbs , j, which is the sum, over the neighboring vertices s of i, of the number of walks of length k − 1 between s and j. ◻
In a strongly regular graph with parameters (n, k, a, c), the number of walks of length 2 between two vertices i and j is
• k if i = j (walk to neighbor and back),
• a if i /= j are adjacent (walk to common neighbor and then to other vertex),
• c if i /= j are not adjacent.
This means that
A2= kI + aA + c(J − I − A) (where J is the all-one matrix). We write this in the form
A2− (a − c)A − (k − c)I = cJ
Note X.13: The adjacency matrix for a strongly regular graph with parameters (v, k, a, a) satisfies – compare to Lemma VIII.8 –
AAT = A2= (k − a)I + aJ
X.2. EIGENVALUES 171 and thus can be interpreted as incidence matrix of a symmetric 2−(v, k, a)-design.
Let v be an eigenvector of A for eigenvalue λ, that is orthogonal to the all-1 vector. (By Note X.10, this can be assumed.) Then
(λ2− (a − c)λ) − (k − c))v = A2v − (a − c)Av − (k − c)v = c Jv = 0 because of the orthogonality of v and the columns of J. This implies that λ2− (a − c)λ) − (k − c) = 0.
Setting ∆ = (a − c)2+ 4(k − c), we thus get that
λ±= (a − c) ±√
∆
2 .
As λ+⋅ λ− = c − k the assumption that c < k implies that λ+and λ−are nonzero with opposite sign.
Let m±be the dimensions of the respective eigenspaces. Then m++ m−= n − 1 and (as the trace of A, the sum of the eigenvalues, is zero) m+λ++ m−λ−= −k.
We solve this system of equations (with signs corresponding) as
m±= ∓(n − 1)λ∓+ k λ+− λ−
and note that
(λ+− λ−)2= (λ++ λ−− 4λ+λ−= (a − c)2+ 4(k − c) = ∆.
This yields the multiplicities
m±= 1
2((n − 1) ∓2k(n − 1)(a − c)√
∆ ) ,
which must be nonnegative integers, imposing a condition on the possible param-eters.
We also note a converse result
Proposition X.14: A connected, regular, graph Γ with exactly three distinct eigen-values is strongly regular.
Proof: Suppose Γ is connected and regular. Then the valency (degree of each vertex) k is an eigenvalue, let λ, µ be the other two. Let A be the adjacency matrix and
M = 1
(k − µ)(k − λ)(A − µI)(A − λI).
Then (using the orthogonality of eigenspaces), M has eigenvalues only 0 or 1 and the eigenspaces of λ and µ in the kernel.
Thus the rank of M is the multiplicity of the eigenvalue k. By Lemma X.10 this multiplicity is one, if Γ is connected.
As the all-one vector is an eigenvector of A (and thus of M) we must have that M = n1J. Thus J is a quadratic polynomial in A and A2is a linear combination of
I, J, A which implies that Γ is strongly regular. ◻
We can interpret this situation also in different language: The matrices A, I, J generate a 3-dimensional C-vector space A that also is a ring under multiplication.
Such a set of matrices is called an algebra, this particlar algebra A is called the Bose-Mesner algebra.
The Krein Bounds
The eigenspaces of A clearly also are eigenspaces for the other two generators I, J of A and thus for all of A. Call them V1,V+and V−and let Eifor i ∈ {1, +, −} be the matrix projecting onto Vi(and mapping the other two eigenspaces to 0). They lie in A, as they can be constructed from A − λI. Vice versa (as any operator can be written as linear combination of the projections to its eigenspaces) we have that A= ⟨E1, E+, E−⟩ (as a vector space or as an algebra).
The Eiare idempotent (that is they satisfy that x ⋅ x = x), they are orthogonal (that is EiE j = 0 if i /= j) and they decompose the identity E1+ E++ E−= I. (Such idempotents are part of a standard analysis of the structure of algebras.)
We now consider a different multiplication ○, called the Hadamard product in which the i, j entry of A ○ B is simply ai , jbi , j. As 0, 1 matrices, A, I and J are idem-potent under the Hadamard product, thus A is closed under this product as well.
This means that
Ei○ Ej= ∑
k
qki jEk
But the Eiare positive definite, and the Hadamard product of positive definite ma-trices it positive definite as well1.
Thus the qki , jmust be positive. Using explicit expressions for the Eiin terms of A, I, J, one can conclude:
Theorem X.15 (Krein bounds): Let Γ be strongly regular such that Γ and its com-plement are connected and let k, r, s be the eigenvalues of Γ. Then
(r + 1)(k + r + 2rs) ≤ (k + r)(s + 1)2, (s + 1)(k + s + 2rs) ≤ (k + s)(r + 1)2.
Association Schemes
The concept of the Bose Mesner algebra can be generalized to higher dimensions:
1This can be shown by observing that A ○ A is a principal submatrix of A ⊗ A.
X.3. MOORE GRAPHS 173