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

New rich infinite families of 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 1 This 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 A 2 = k · I + λ · A + µ · ( J − I − A ) , or equivalently, A 2 + ( 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 . A 2 = tI + λ A + µ ( J − I − A ) . ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 4 / 24

Directed strongly regular graphs µ λ t k − t k − t x u w u w 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 , ¯ t , ¯ µ ) with adjacency matrix ¯ k , ¯ λ, ¯ A = J − I − A , where ¯ k = n − k + 1 ¯ = n − 2 k + t − 1 t ¯ λ = n − 2 k + µ − 2 µ ¯ = n − 2 k + 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 Γ T is also a DSRG with the same parameter set. Definition We say that two DSRGs Γ 1 and Γ 2 are equivalent, if Γ 1 ∼ = Γ 2 , or Γ 1 ∼ = Γ T 2 , or Γ 1 ∼ = ¯ Γ 2 , or Γ 1 ∼ = ¯ Γ T 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 ) = d 2 d | (2 k − ( µ − λ )( n − 1)) 2 k − ( µ − λ )( n − 1) ≡ n − 1 (mod 2) d � 2 k − ( µ − λ )( n − 1) � � � � ≤ n − 1 . � � d � ˇ Stefan Gy¨ urki (STU Bratislava) Directed strongly regular graphs June 2014 9 / 24

Directed strongly regular graphs Further necessary conditions 0 ≤ λ < t < k 0 < 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 Γ 1 of order 2 n 2 , whose vertex set is V (Γ 1 ) = { 1 , 2 , . . . , n } × { 1 , 2 , . . . , n } × Z 2 . The set D (Γ 1 ) of darts is defined as follows: ( x , y , i ) �→ ( z , y , i ) for all i ∈ Z 2 , x , y , z ∈ { 1 , 2 , . . . , n } , x � = z ; ( x , y , i ) �→ ( x , z , i ) for all i ∈ Z 2 , 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. Γ 1 is a DSRG with parameter set (2 n 2 , 3 n − 2 , 2 n − 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 Γ 2 of order 3 n 2 , whose vertex set is V (Γ 2 ) = { 1 , 2 , . . . , n } × { 1 , 2 , . . . , n } × Z 3 . The set D (Γ 2 ) of darts is defined as follows: ( x , y , i ) �→ ( z , y , i ) for all i ∈ Z 3 , x , y , z ∈ { 1 , 2 , . . . , n } , x � = z ; ( x , y , i ) �→ ( x , z , i ) for all i ∈ Z 3 , x , y , z ∈ { 1 , 2 , . . . , n } , y � = z ; ( x , y , i ) �→ ( xy , z , i + 1) for all i ∈ Z 3 , and z ∈ { 1 , 2 , . . . , n } . ( x , y , i ) �→ ( z , yx , i − 1) for all i ∈ Z 3 , and z ∈ { 1 , 2 , . . . , n } . Theorem 2. Γ 2 is a DSRG with parameter set (3 n 2 , 4 n − 2 , 2 n , 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 Γ 3 of order 2 n 2 , whose vertex set is V (Γ 3 ) = { 1 , 2 , . . . , n } × { 1 , 2 , . . . , n } × Z 2 . The set D (Γ 3 ) of darts is defined as follows: ( x , y , i ) �→ ( z , y , i ) for all i ∈ Z 2 , x , y , z ∈ { 1 , 2 , . . . , n } , x � = z ; ( x , y , i ) �→ ( x , z , i ) for all i ∈ Z 2 , 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. Γ 3 is a DSRG with parameter set (2 n 2 , 4 n − 2 , 2 n + 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 Γ 4 of order 3 n 2 , whose vertex set is V (Γ 4 ) = { 1 , 2 , . . . , n } × { 1 , 2 , . . . , n } × Z 3 . The set D (Γ 4 ) of darts is defined as follows: ( x , y , i ) �→ ( z , y , i ) for all i ∈ Z 3 , x , y , z ∈ { 1 , 2 , . . . , n } , x � = z ; ( x , y , i ) �→ ( x , z , i ) for all i ∈ Z 3 , 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. Γ 4 is a DSRG with parameter set (3 n 2 , 6 n − 2 , 2 n + 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 Γ 1 for the group case Theorem 5. If Γ 1 from Construction 1 is arising from a group K , then for its full group G of automorphisms holds: = ( K 2 ⋊ Aut ( K )) ⋊ S 2 . G ∼ 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