New rich in nite families · where I is the identity matrix and J the all-one matrix. Stefan Gyurki...

New rich infinite familiesof directed strongly regular graphs 1

Stefan Gyurki(joint work with M. Klin)

Slovak University of Technology in Bratislava, Slovakia

Modern Trends in Algebraic Graph TheoryJune 2014

1This research was supported at Matej Bel University (Slovakia) by the EuropeanSocial Fund, ITMS code: 26110230082.

Stefan Gyurki (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 anyu, 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 .

Stefan Gyurki (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),

or equivalently,

A2 + (µ− λ) · A− (k − µ) · I = µ · J,

where I is the identity matrix and J the all-one matrix.

Stefan Gyurki (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 haveconstant 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 withparameters (n, k , t, λ, µ) if there exist constants λ and µ such that thenumbers 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).

Stefan Gyurki (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.

Stefan Gyurki (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).

Stefan Gyurki (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.

Stefan Gyurki (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 sameparameter set.

Definition

We say that two DSRGs Γ1 and Γ2 are equivalent, if Γ1∼= Γ2, or Γ1

∼= ΓT2 ,

or Γ1∼= Γ2, or Γ1

∼= ΓT2 ; otherwise they are called non-equivalent.

Stefan Gyurki (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 somepositive 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.

Stefan Gyurki (STU Bratislava) Directed strongly regular graphs June 2014 9 / 24

Directed strongly regular graphs

Further necessary conditions

0 ≤ λ < t < k0 < µ ≤ t < k

−2(k − t − 1) ≤ µ− λ ≤ 2(k − t).

Stefan Gyurki (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 andS. Hobart: http://homepages.cwi.nl/~aeb/math/dsrg

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Combinatorial structures

Definition

A Latin square of order n is an n × n array with n different entries, suchthat each entry occurs exactly once in any row and in any column of thearray.

A quasigroup is a set Q with a binary operation “·” such that for alla, 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 propertye · x = x · e = x for every x ∈ L.

Stefan Gyurki (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 isV (Γ1) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z2.The set D(Γ1) of darts is defined as follows:

(x , y , i) 7→ (z , y , i) for all i ∈ Z2, x , y , z ∈ {1, 2, . . . , n}, x 6= z ;

(x , y , i) 7→ (x , z , i) for all i ∈ Z2, x , y , z ∈ {1, 2, . . . , n}, y 6= z ;

(x , y , 0) 7→ (xy , z , 1) for all z ∈ {1, 2, . . . , n}.(x , y , 1) 7→ (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).

Stefan Gyurki (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 isV (Γ2) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z3.The set D(Γ2) of darts is defined as follows:

(x , y , i) 7→ (z , y , i) for all i ∈ Z3, x , y , z ∈ {1, 2, . . . , n}, x 6= z ;

(x , y , i) 7→ (x , z , i) for all i ∈ Z3, x , y , z ∈ {1, 2, . . . , n}, y 6= z ;

(x , y , i) 7→ (xy , z , i + 1) for all i ∈ Z3, and z ∈ {1, 2, . . . , n}.(x , y , i) 7→ (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).

Stefan Gyurki (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-identityelement of L.Define a digraph Γ3 of order 2n2, whose vertex set isV (Γ3) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z2.The set D(Γ3) of darts is defined as follows:

(x , y , i) 7→ (z , y , i) for all i ∈ Z2, x , y , z ∈ {1, 2, . . . , n}, x 6= z ;

(x , y , i) 7→ (x , z , i) for all i ∈ Z2, x , y , z ∈ {1, 2, . . . , n}, y 6= z ;

(x , y , 0) 7→ (xy , z , 1) for all z ∈ {1, 2, . . . , n}.(x , y , 1) 7→ (z , yx , 0) for all z ∈ {1, 2, . . . , n}.(x , y , 0) 7→ (c(xy), z , 1) for all z ∈ {1, 2, . . . , n}.(x , y , 1) 7→ (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).

Stefan Gyurki (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-identityelement of L.Define a digraph Γ4 of order 3n2, whose vertex set isV (Γ4) = {1, 2, . . . , n} × {1, 2, . . . , n} × Z3.The set D(Γ4) of darts is defined as follows:

(x , y , i) 7→ (z , y , i) for all i ∈ Z3, x , y , z ∈ {1, 2, . . . , n}, x 6= z ;

(x , y , i) 7→ (x , z , i) for all i ∈ Z3, x , y , z ∈ {1, 2, . . . , n}, y 6= z ;

(x , y , i) 7→ (xy , z , i + 1) for all z ∈ {1, 2, . . . , n}.(x , y , i) 7→ (z , yx , i − 1) for all z ∈ {1, 2, . . . , n}.(x , y , i) 7→ (c(xy), z , i + 1) for all z ∈ {1, 2, . . . , n}.(x , y , i) 7→ (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).

Stefan Gyurki (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.

Stefan Gyurki (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 groupG of automorphisms holds:

G ∼= (K 2 oAut(K )) o S2.

Remark

The proof follows from the classical results about the automorphismgroups of 3-nets, corresponding to group Latin squares (see e.g. survey ofHeinze and Klin).

Stefan Gyurki (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 thatfor all a, b, c ∈ Q1: (a · b = c) ⇐⇒ (af ◦ bf = c f ).

Two Latin squares L1, L2 represented as n × n-arrays are isotopic,if there exist three permutations h1, h2, h3 ∈ Sn such that the actionh1 on rows, h2 on columns, h3 on symbols of L1 brings L1 to L2.

Stefan Gyurki (STU Bratislava) Directed strongly regular graphs June 2014 19 / 24

Numbers of DSRGs

Considering loops of order n, our four constructions give the followingnumber of non-isomorphic DSRGs:

n ISOTC ISOMC Constr.1. Constr.2. Constr.3 Constr.4

3 1 1 1 1 1 14 2 2 2 2 2 25 2 6 3 6 9 106 22 109 38 109 341 365

Table: Numbers of different combinatorial objectsISOTC = nr. of isotopy classes of loopsISOMC = nr. of isomorphism classes of loops

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Conjectures

Conjecture 1.

The number of non-isomorphic DSRG’s from our constructions growsexponentially over n.

Conjecture 2.

The number of non-isomorphic DSRG’s from our Construction 2 is equalto the number of isomorphism classes of loops.

Stefan Gyurki (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. Vojtechovsky)

Stefan Gyurki (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 etal., Algorithmic Algebraic Combinatorics and Grobner Bases, SpringerVerlag, Berlin Heidelberg, 2009, 3–65.

M. Klin, A. Munemasa, M. Muzychuk, P.-H. Zieschang, Directedstrongly regular graphs obtained from coherent algebras, Lin. Alg.Appl. 377 (2004) 83–109.

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

Stefan Gyurki (STU Bratislava) Directed strongly regular graphs June 2014 23 / 24

Thank you

Thank you for your attention.

Stefan Gyurki (STU Bratislava) Directed strongly regular graphs June 2014 24 / 24