New rich infinite families of directed strongly regular graphs 1 - - PowerPoint PPT Presentation

New rich infinite families

• f directed strongly regular graphs 1

ˇ Stefan Gy¨ urki (joint work with M. Klin)

Slovak University of Technology in Bratislava, Slovakia

Modern Trends in Algebraic Graph Theory June 2014

1This research was supported at Matej Bel University (Slovakia) by the European

Social Fund, ITMS code: 26110230082.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 1 / 24

Strongly regular graphs

Definition A simple graph Γ = (V , E) is called strongly regular with parameters (n, k, λ, µ), if |V | = n and there exist constants k, λ, µ such that for any u, v ∈ V the number of uv-walks of length 2 is

1 k, if u = v, 2 λ, if (u, v) ∈ E, 3 µ, if (u, v) /

∈ E.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 2 / 24

Strongly regular graphs

Let A = A(Γ) denote the adjacency matrix of Γ. Then A2 = k · I + λ · A + µ · (J − I − A),

• r equivalently,

A2 + (µ − λ) · A − (k − µ) · I = µ · J, where I is the identity matrix and J the all-one matrix.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 3 / 24

Directed strongly regular graphs

Definition (Duval, 1988) Let Γ = (V , D) be a directed graph, |V | = n, in which vertices have constant in- and out-valency k, but now only t edges being undirected (0 < t < k). We say that Γ is a directed strongly regular graph with parameters (n, k, t, λ, µ) if there exist constants λ and µ such that the numbers of uw-paths of length 2 are

1 t, if u = w; 2 λ, if (u, w) ∈ D; 3 µ, if (u, w) /

∈ D. A2 = tI + λA + µ(J − I − A).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 4 / 24

Directed strongly regular graphs

x u w u w t k − t k − t λ µ Figure: Locally.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 5 / 24

Directed strongly regular graphs

Figure: The smallest DSRG.

The parameter set is (6, 2, 1, 0, 1).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 6 / 24

Directed strongly regular graphs

Proposition (Duval, 1988) If Γ is a DSRG with parameter set (n, k, t, λ, µ) and adjacency matrix A, then the complementary graph ¯ Γ is a DSRG with parameter set (n, ¯ k,¯ t, ¯ λ, ¯ µ) with adjacency matrix ¯ A = J − I − A, where ¯ k = n − k + 1 ¯ t = n − 2k + t − 1 ¯ λ = n − 2k + µ − 2 ¯ µ = n − 2k + λ.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 7 / 24

Directed strongly regular graphs

Proposition (Ch. Pech, 1997) [Presented in KMMZ] Let Γ be a DSRG. Then its reverse ΓT is also a DSRG with the same parameter set. Definition We say that two DSRGs Γ1 and Γ2 are equivalent, if Γ1 ∼ = Γ2, or Γ1 ∼ = ΓT

2 ,

• r Γ1 ∼

= ¯ Γ2, or Γ1 ∼ = ¯ ΓT

2 ; otherwise they are called non-equivalent.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 8 / 24

Directed strongly regular graphs

Duval’s main theorem Let Γ be a DSRG with parameters (n, k, t, λ, µ). Then there exists some positive integer d for which the following requirements are satisfied: k(k + (µ − λ)) = t + (n − 1)µ (µ − λ)2 + 4(t − µ) = d2 d | (2k − (µ − λ)(n − 1)) 2k − (µ − λ)(n − 1) d ≡ n − 1 (mod 2)

• 2k − (µ − λ)(n − 1)

d

• ≤ n − 1.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 9 / 24

Directed strongly regular graphs

Further necessary conditions ≤ λ < t < k < µ ≤ t < k −2(k − t − 1) ≤ µ − λ ≤ 2(k − t).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 10 / 24

Directed strongly regular graphs

Usually, the main goals concerning DSRG’s are:

1 To find a DSRG realizing a “new” parameter set. 2 To prove a non-existence result. 3 To find an infinite family of DSRG’s.

The most important data are collected on the webpage of A. Brouwer and

• S. Hobart: http://homepages.cwi.nl/~aeb/math/dsrg

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 11 / 24

Combinatorial structures

Definition A Latin square of order n is an n × n array with n different entries, such that each entry occurs exactly once in any row and in any column of the array. A quasigroup is a set Q with a binary operation “·” such that for all a, b ∈ Q the equations a · x = b and y · a = b have a unique solution in Q. A loop L is a quasigroup with an identity element e ∈ L with the property e · x = x · e = x for every x ∈ L.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 12 / 24

Construction 1.

Let (Q, ·) be an arbitrary quasigroup of order n ≥ 2. Define a digraph Γ1 of order 2n2, whose vertex set is V (Γ1) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z2. The set D(Γ1) of darts is defined as follows: (x, y, i) → (z, y, i) for all i ∈ Z2, x, y, z ∈ {1, 2, . . . , n}, x = z; (x, y, i) → (x, z, i) for all i ∈ Z2, x, y, z ∈ {1, 2, . . . , n}, y = z; (x, y, 0) → (xy, z, 1) for all z ∈ {1, 2, . . . , n}. (x, y, 1) → (z, yx, 0) for all z ∈ {1, 2, . . . , n}. Theorem 1. Γ1 is a DSRG with parameter set (2n2, 3n − 2, 2n − 1, n − 1, 3).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 13 / 24

Construction 2.

Let (Q, ·) be an arbitrary quasigroup of order n ≥ 2. Define a digraph Γ2 of order 3n2, whose vertex set is V (Γ2) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z3. The set D(Γ2) of darts is defined as follows: (x, y, i) → (z, y, i) for all i ∈ Z3, x, y, z ∈ {1, 2, . . . , n}, x = z; (x, y, i) → (x, z, i) for all i ∈ Z3, x, y, z ∈ {1, 2, . . . , n}, y = z; (x, y, i) → (xy, z, i + 1) for all i ∈ Z3, and z ∈ {1, 2, . . . , n}. (x, y, i) → (z, yx, i − 1) for all i ∈ Z3, and z ∈ {1, 2, . . . , n}. Theorem 2. Γ2 is a DSRG with parameter set (3n2, 4n − 2, 2n, n, 4).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 14 / 24

Construction 3.

Let (L, ·) be an arbitrary loop of order n ≥ 2, and c any non-identity element of L. Define a digraph Γ3 of order 2n2, whose vertex set is V (Γ3) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z2. The set D(Γ3) of darts is defined as follows: (x, y, i) → (z, y, i) for all i ∈ Z2, x, y, z ∈ {1, 2, . . . , n}, x = z; (x, y, i) → (x, z, i) for all i ∈ Z2, x, y, z ∈ {1, 2, . . . , n}, y = z; (x, y, 0) → (xy, z, 1) for all z ∈ {1, 2, . . . , n}. (x, y, 1) → (z, yx, 0) for all z ∈ {1, 2, . . . , n}. (x, y, 0) → (c(xy), z, 1) for all z ∈ {1, 2, . . . , n}. (x, y, 1) → (z, (yx)c, 0) for all z ∈ {1, 2, . . . , n}. Theorem 3. Γ3 is a DSRG with parameter set (2n2, 4n − 2, 2n + 2, n + 2, 6).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 15 / 24

Construction 4.

Let (L, ·) be an arbitrary loop of order n ≥ 2, and c any non-identity element of L. Define a digraph Γ4 of order 3n2, whose vertex set is V (Γ4) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z3. The set D(Γ4) of darts is defined as follows: (x, y, i) → (z, y, i) for all i ∈ Z3, x, y, z ∈ {1, 2, . . . , n}, x = z; (x, y, i) → (x, z, i) for all i ∈ Z3, x, y, z ∈ {1, 2, . . . , n}, y = z; (x, y, i) → (xy, z, i + 1) for all z ∈ {1, 2, . . . , n}. (x, y, i) → (z, yx, i − 1) for all z ∈ {1, 2, . . . , n}. (x, y, i) → (c(xy), z, i + 1) for all z ∈ {1, 2, . . . , n}. (x, y, i) → (z, (yx)c, i − 1) for all z ∈ {1, 2, . . . , n}. Theorem 4. Γ4 is a DSRG with parameter set (3n2, 6n − 2, 2n + 6, n + 6, 10).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 16 / 24

Proof of Theorems 1-4. (outline)

existence of k and t; existence of λ and µ:

counting over darts and non-darts; various types of directed paths of length 2; uniqueness of solutions of equations x · a = b and a · y = b.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 17 / 24

Automorphism group of Γ1 for the group case

Theorem 5. If Γ1 from Construction 1 is arising from a group K, then for its full group G of automorphisms holds: G ∼ = (K 2 ⋊ Aut(K)) ⋊ S2. Remark The proof follows from the classical results about the automorphism groups of 3-nets, corresponding to group Latin squares (see e.g. survey of Heinze and Klin).

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 18 / 24

Isotopism and isomorphism of quasigroups

Two quasigroups (Q1, ·) and (Q2, ◦) are isomorphic, if there exists a bijection f from Q1 to Q2 such that for all a, b, c ∈ Q1: (a · b = c) ⇐ ⇒ (af ◦ bf = cf ). Two Latin squares L1, L2 represented as n × n-arrays are isotopic, if there exist three permutations h1, h2, h3 ∈ Sn such that the action h1 on rows, h2 on columns, h3 on symbols of L1 brings L1 to L2.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 19 / 24

Numbers of DSRGs

Considering loops of order n, our four constructions give the following number of non-isomorphic DSRGs: n ISOTC ISOMC Constr.1. Constr.2. Constr.3 Constr.4 3 1 1 1 1 1 1 4 2 2 2 2 2 2 5 2 6 3 6 9 10 6 22 109 38 109 341 365

Table: Numbers of different combinatorial objects ISOTC = nr. of isotopy classes of loops ISOMC = nr. of isomorphism classes of loops

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 20 / 24

Conjectures

Conjecture 1. The number of non-isomorphic DSRG’s from our constructions grows exponentially over n. Conjecture 2. The number of non-isomorphic DSRG’s from our Construction 2 is equal to the number of isomorphism classes of loops.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 21 / 24

Computer tools

GAP (Group Algorithm and Programming) COCO II (unpublished version by S. Reichard) GRAPE (L.H. Soicher) nauty (B.D. McKay) LOOPS (G.P. Nagy, P. Vojtˇ echovsk´ y)

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 22 / 24

References

A.M. Duval, A directed version of strongly regular graphs, J. Comb.

• Th. A 47(1988), 71–100.
• A. Heinze, M. Klin, Loops, Latin squares and strongly regular graphs:

An algorithmic approach via algebraic combinatorics, in: M. Klin et al., Algorithmic Algebraic Combinatorics and Gr¨

• bner Bases, Springer

Verlag, Berlin Heidelberg, 2009, 3–65.

• M. Klin, A. Munemasa, M. Muzychuk, P.-H. Zieschang, Directed

strongly regular graphs obtained from coherent algebras, Lin. Alg.

• Appl. 377 (2004) 83–109.

http://homepages.cwi.nl/~aeb/math/dsrg

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 23 / 24

Thank you

Thank you for your attention.

ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 24 / 24