Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs. Sung Y. Song Iowa State University

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1 Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs Sung Y. Song Iowa State University

2 Notation K: one of the fields R or C X: a nonempty finite set Mat X (K): the K-algebra consisting of the matrices over K, and whose rows and columns are indexed by X K X : the K-vector space of column vectors over K, and whose coordinates are indexed by X Observe Mat X (K) acts on K X by left multiplication.

3 We endow R X with the dot product u, v = u t v (u, v R X ), and we endow C X with the Hermitean dot product u, v = u t v (u, v C X ). For each x in X, let ˆx denote the vector in K X with a 1 in coordinate x, and zeros in all other coordinates. Observe that {ˆx : x X} is an orthonormal basis for K X.

4 Let Γ = (X, R) denote a graph. The binary relation on X of being connected by a path is an equivalence relation; the equivalence classes are known as the connected components of Γ. Γ is said to be connected whenever X is a connected component. Assume Γ is connected, and pick x, y X. By the distance from x to y, we mean the scalar (x, y) = min{l : path of length l from x to y}. By the diameter of Γ, we mean the scalar D := max{ (x, y) : x, y X}.

5 Let Γ be a finite simple (undirected) connected graph. The adjacency matrix for Γ is the matrix A Mat X (C) with xy entry 1 if {x, y} R A xy = ( x, y X). 0 if {x, y} R The Bose-Mesner algebra of Γ over K, is the subalgebra M of Mat X (K) generated by the adjacency matrix A of Γ. Observe that: the dimension of M as a K-vector space = the degree of the minimal polynomial of A = the number of distinct eigenvalues of Γ.

6 Lemma. Let Γ be a connected graph with diameter D. Let M denote the Bose-Mesner algebra of Γ over K. (i) The dimension of M as a K-vector space is at least D + 1. (ii) Γ has at least D + 1 distinct eigenvalues.

7 Let Γ = (X, R) denote a connected graph with diameter D. For each integer i (0 i D), let A i denote the matrix in Mat X (C) with x, y entry 1 if (x, y) = i (A i ) xy = ( x, y X). 0 if (x, y) i We call A i the ith distance matrix for Γ.

8 Let Γ = (X, R) denote a connected graph with diameter D, and distance matrices A 0, A 1,...,A D. (i) A 0, A 1,...,A D are linearly independent. (ii) For all x X, and for all integers i (0 i D), A iˆx = y X (x,y)=i ŷ.

9 Lemma. Let Γ = (X, R) be a connected graph with diameter D, and distance matrices A 0, A 1,...,A D. (i) A 0 = I. (ii) A 1 = A. (iii) A 0 + A A D = J (the all 1s matrix). (iv) A t i = A i (0 i D).

10 Let Γ = (X, R) denote a connected graph with diameter D. Γ is said to be distance-regular whenever for all integers i, 0 i D and for all x, y X at distance (x, y) = i, the scalars c i := {z X : (x, z) = i 1, (y, z) = 1}, a i := {z X : (x, z) = i, (y, z) = 1}, b i := {z X : (x, z) = i + 1, (y, z) = 1} are constants that are independent of x and y. We refer to the c i, a i, b i as the intersection numbers of Γ.

11 Lemma Let Γ = (X, R) denote a distance-regular graph with diameter D. (a) c 0 = 0, a 0 = 0, b D = 0, c 1 = 1, and that c i > 0 (1 i D), b i > 0 (0 i D 1). (b) Γ is regular with valency k := b 0. (c) k = c i + a i + b i (0 i D).

12 Distance-regular graphs with diameter D The Hamming graph H(D, N), (N 2). X = the set of D-tuples of elements from the set {1, 2,...,N}, R = {xy : x, y X, x and y differ in exactly one coordinate}. Here c i = (N 1)i, b i = (D i)(n 1) (0 i D). H(D, 2) is called the D-cube.

13 The Johnson graph J(D, N), (N 2D). X = the set of D-element subsets of {1, 2,...,N}, R = {xy : x, y X, x y = D 1}. Here c i = i 2, b i = (D i)(n D i) (0 i D).

14 The q-johnson graph J q (D, N), (N 2D). Let V denote an N-dimensional vector space over a finite field GF(q). X = the set of D-dimensional subspaces of V. R = {xy : x, y X, dim(x y) = D 1}. Here c i = ( q i 1 ) 2 (0 i D), q 1 b i = q2i+1 (q D i 1)(q N D i 1) (q 1) 2 (0 i D).

15 Lemma. Let Γ = (X, R) be a distance-regular graph with diameter D. Then the distance matrices satisfy AA i = c i+1 A i+1 + a i A i + b i 1 A i 1 (1 i D), where A D+1 = 0, and for convenience c D+1 = 1.

16 Lemma. Let Γ = (X, R) denote a distance-regular graph with diameter D. Let M denote the Bose-Mesner algebra of Γ over K. (i) The distance matrices A 0, A 1,...,A D form a basis for M. (ii) The dimension of M as a K-vector space equals D + 1. (iii) Γ has exactly D + 1 distinct eigenvalues.

17 How to compute the eigenvalues of a distance-regular graph from the intersection numbers? Def. Let Γ = (X, R) be a distance-regular graph with diameter D. We define the polynomials f 0, f 1,...,f D+1 C[λ] by f 0 = 1, f 1 = λ, λf i = c i+1 f i+1 + a i f i + b i 1 f i 1 (1 i D), where for convenience we set c D+1 = 1.

18 Lemma. Let the polynomials f 0, f 1,...,f D+1 be as in the above definition. (i) deg f i = i (0 i D + 1). (ii) The coefficient of λ i in f i is (c 1 c 2 c i ) 1 (0 i D + 1). (iii) f i (A) = A i (0 i D + 1). In particular, f D+1 (A) = 0. (iv) The distinct eigenvalues of Γ are precisely the zeros of f D+1.

19 We now wish to expand our notion of an intersection number. Let Γ = (X, R) denote a distance-regular graph with diameter D. Since the distance matrices A 0, A 1,...,A D are a basis for the Bose-Mesner algebra, there exist scalars p h ij C (0 h, i, j D) such that D A i A j = (0 i, j D). h=0 Since A i and A j commute, p h ija h p h ij = p h ji (0 h, i, j D).

20 We find that for all integers h, i, j (0 h, i, j D), and for all x, y X such that (x, y) = h, p h ij = {z X (x, z) = i, (y, z) = j}. In particular, each p h ij definitions, we see that is a nonnegative integer. Comparing the c i = p i i 1,1 a i = p i i1 (1 i D), (0 i D), b i = p i i+1,1 (0 i D 1). In view of this, we often refer to any p h ij number of Γ. as an intersection

21 It turns out that all of the p h ij b 0, b 1,...,b D 1, c 1, c 2,...,c D. can be computed from Set k i := p 0 ii (0 i D). With h = 0, i = j, x = y, we find that for all integers i (0 i D), and for all x X, k i = {z X : (x, z) = i}. In particular, and k 0 = 1, k 1 = k, k i 0 (0 i D). We refer to k i as the ith valency of Γ.

22 Lemma. Let Γ = (X, R) be a distance-regular graph with diameter D. Then (i) k h p h ij = k ip i hj = k jp j ih (0 h, i, j D). (ii) k i 1 b i 1 = k i c i (iii) (1 i D). k i = b 0b 1 b i 1 c 1 c 2 c i (0 i D).

23 Lemma. Let Γ = (X, R) denote a distance-regular graph with diameter D. Then for all integers h, i, j (0 h, i, j D), (i) p h ij (ii) p h ij = 0 if one of h, i, j is greater than the sum of the other two; 0 if one of h, i, j is equal to the sum of the other two.

24 In the sequel, we hope to continue to study on distance-regular graphs. However, (considering the schedule having been booked up for the semester,) next time it is illuminating to consider a slightly more general object called a commutative association scheme. For these schemes one still has intersection numbers p h ij ; the distance regular graphs correspond to the case where the intersection numbers satisfy (i) and (ii) in the last lemma.