Directed Strongly Regular Graphs. Leif K. Jørgensen Aalborg University Denmark

Transcription

1 Directed Strongly Regular Graphs Leif K. Jørgensen Aalborg University Denmark

2 A. Duval 1988: A directed strongly regular graph with parameters v, k, t, λ, µ is a directed graph with the following properties: 1. every vertex has indegree and outdegree k 2. for every pair of vertices x, y the number of vertices z so that x z y is t if x = y, λ if x y, µ otherwise.

3 A directed graph with vertices x 1,..., x v the adjacency matrix is the v v matrix A with A ij = 1 if x i x j 0 otherwise. A directed graph is strongly regular if its adjacency matrix A satisfies 1. AJ = JA = kj 2. A 2 = ti + λa + µ(j I A) For µ > 0 the first condition follows from the second condition: A 2 = ti + λa + µ(j I A) J = 1 µ A2 + µ λ) µ A + µ t µ I AJ = JA

4 For t = k the graph is undirected. It is strongly regular. This graph is a relation of a symmetric association scheme with 2 classes. For t = 0 we have µ = λ + 1, k = 2λ + 1, v = 2k + 1 = 4λ + 3. The graph is a doubly regular tournament. This graph is a relation of a non-symmetric association scheme with 2 classes.

5 Directed Moore graphs A directed graph with diameter 2 and maximum outdegree k has at most 1 + k + k 2 vertices. If it has exactly 1+k +k 2 vertices then it is a directed strongly regular graph with (v, k, t, λ, µ) = (1 + k + k 2, k, 0, 0, 1). But directed Moore graph do not exist. (Plesnik and Znam 1974).

6 A directed strongly regular graph with (v, k, t, λ, µ) = (k+k 2, k, 1, 0, 1) is an almost Moore directed graph. Theorem (Gimbert 1999) For each k there is a unique directed strongly regular graph with (v, k, t, λ, µ) = (k + k 2, k, 1, 0, 1). It is the line graph of a complete directed graph of order k + 1. The adjacency matrix of this directed graph has rank k + 1.

7 Theorem (Duval 1988) Suppose that there exists a directed strongly regular graph with parameters v, k, µ, λ, t, 0 < t < k. Then the parameters satisfy k(k + (µ λ)) = t + (v 1)µ 0 λ < t, 0 < µ t, 2(k t 1) µ λ 2(k t). The eigenvalues of the adjacency matrix are k > ρ = 1 2 ( (µ λ) + d) > σ = 1 ( (µ λ) d), 2 for some positive integer d, where d 2 = (µ λ) 2 + 4(t µ). The multiplicities are k + σ(n 1) 1,, ρ σ respectively. k + ρ(n 1), ρ σ If these conditions are satisfied then we say that the parameters are feasible. Feasible does not necessarily mean that a graph exists.

8 Directed strongly regular graphs with eigenvalue 0. Equivalent condition: t = µ. For (undirected) strongly regular graphs this is a complete multipartite graph. Theorem (J. 2005) Let r be a positive integer and let q be a rational number. Then there exists a feasible parameter set (v, k, µ, λ, t) with rank r and with k v = q if and only if 1 r q 2r 3. 2r Proof of if: If 1 r q 2r 3 2r and q = a b and b then (v, k, t, λ, µ) = for integers a ((r 1)b 2 m, (r 1)abm, ra 2 m, (ar b)am, ra 2 m) is a feasible parameter set with rank r, for every positive integer m.

9 We consider matrices A such that every entry of A is either 0 or 1 A is an n n matrix, for some n A has exactly k 1 s in each column and exactly k 1 s in each row, for some k. For a given number r, let B r be the set of all values of n k for which there exists a matrix with the above properties and with rank r.

10 Example The following matrix shows that 4 9 B

11 Theorem (J. 2005) Every set B r is finite, 2 r 1 1 B r < 2 r2. B 1 = {1}, B 2 = { 1 2 }, B 3 = { 1 3, 1 2, 2 3 }, B 4 = { 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4 }, B 5 = { 1 5, 1 4, 2 7, 1 3, 3 8, 2 5, 3 7, 4 9, 1 2, 5 9, 4 7, 3 5, 5 8, 2 3, 5 7, 3 4, 4 5 }, B 6 = 45 B 7 = 147

12 Fiedler, Klin and Muzychuk (2002) proved that is not realizable (rank 4). (16, 6, 3, 1, 3) Theorem (J. 2003) If rank = 3 then either (v, k, t, λ, µ) = (6m, 2m, m, 0, m) or (v, k, t, λ, µ) = (8m, 4m, 3m, m, 3m) and if rank = 4 then either (v, k, t, λ, µ) = (6m, 3m, 2m, m, 2m) or (v, k, t, λ, µ) = (12m, 3m, m, 0, m), for some positive integer m. For rank 3 this was proved independently by Godsil, Hobart and Martin 2007.

13 Let B = J I A. Let A be the vectorspace spanned by {I, A, B}. Then A is closed under ordinary multiplication. A is also closed under Hadamard (entrywise) multiplication: I I = I, A A = A, B B = B, I A = I B = A I = A B = B I = B A = 0. If A if closed under transposition then (t = k or t = 0 and) we have a Bose-Mesner algebra.

14 Since A has a diagonalization, there exist projections E k, E ρ and E σ on the corresponding eigenspace with nullspace spanned by the other two eigenspaces. I = E k + E ρ + E σ A = ke k + ρe ρ + σe σ. B = (n k 1)E k (1 + ρ)e ρ (1 + σ)e σ. From these equations we find that E k = 1 n I + 1 n A + 1 n B, E ρ = k + nσ σ n(σ ρ) I + k n σ n(σ ρ) A + E σ = k + nρ ρ n(ρ σ) I + k n ρ n(ρ σ) A + k σ n(σ ρ) B, k ρ n(ρ σ) B.

15 There exist rational numbers qθφ ω, for θ, φ, ω {k, ρ, σ} such that E θ E φ = 1 v (qk θφ E k + q ρ θφ E ρ + q σ θφ E σ). In particular, for {θ, φ} = {ρ, σ}: q θ θθ = n(θ + φ2 ) 2(θ φ)(k φ) (θ φ) 2, q k θθ = nk k2 + 2kφ + nφ 2 φ 2 (θ φ) 2, q φ θθ = nφ(1 + φ) (θ φ) 2.

16 Lemma (Neumaier 1981) Let M be any matrix of rank r. Then M M has rank at most 2 1 r(r + 1). Theorem (J. 2003) Let G be a directed strongly regular graph. Let m be the multiplicity of an eigenvalue θ k. Suppose that the eigenvalue φ of A different from k and θ satisfies φ {0, 1}. Then v 1 2m(m + 3). If qθθ θ 0 then v 1 2m(m + 1).

17 Theorem (J. 2001) For every two natural numbers µ and k where µ divides k 1 there exists a directed strongly regular graph with v = (k + 1) k 1 µ t = µ + 1 λ = µ. The vertices are integers modulo v. There is a directed edge x y iff x + ky {1,..., k} mod v. The adjacency matrix satisfies A 2 = I + µj.

18 Let G be a group and let S G. Then the Cayley of G with connection set S is the graph with vertex set G and an edge x y iff x 1 y S. For k = 2µ + 1 the above directed strongly regular graph is a Cayley of the dihedral group of order v = 4µ + 4. These Cayley graphs were first found by Hobart and Shaw 1999.

19 Theorem (J. 2001) A directed strongly regular graph with 0 < t < k can not be a Cayley of an abelian group. Proof (Savchenko) Lemma If A is adjacency matrix of a Cayley graph of an abelian group then AA t = A t A. Suppose that A is the adjacency matrix of a directed strongly regular graph which is a Cayley graph of an abelian group. Then A is normal. The eigenvalues of A are real, as t > 0. Thus A is self-adjoint, and so A = A t. I.e., the is undirected, t = k.

20 Theorem (J. 2003) The parameters of a directed strongly regular graph satisfy (k t)(µ (k t)) λt. For a fixed vertex x let l denote the number of edges y z, where x y, y x and x y. Then (k t)(µ (k t)) l λt.