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Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Some Families of Directed Strongly Regular Graphs Obtained from Certain Finite Incidence Structures Oktay Olmez Department of Mathematics Iowa State University 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing May 12, 2011 1 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Overview 1 Directed Strongly Regular Graphs 2 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Overview 1 Directed Strongly Regular Graphs 2 Combinatorial Designs 3 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Overview 1 Directed Strongly Regular Graphs 2 Combinatorial Designs 3 Directed Strongly Regular Graphs Obtained from Affine Planes 4 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Overview 1 Directed Strongly Regular Graphs 2 Combinatorial Designs 3 Directed Strongly Regular Graphs Obtained from Affine Planes 4 Directed Strongly Regular Graphs Obtained from Tactical Configurations 5 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Directed strongly regular graphs(DSRG) Definition A loopless directed graph D with v vertices is called directed strongly regular graph with parameters ( v , k , t , λ, µ ) if and only if D satisfies the following conditions: 6 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Directed strongly regular graphs(DSRG) Definition A loopless directed graph D with v vertices is called directed strongly regular graph with parameters ( v , k , t , λ, µ ) if and only if D satisfies the following conditions: - Every vertex has in-degree and out-degree k . 7 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Directed strongly regular graphs(DSRG) Definition A loopless directed graph D with v vertices is called directed strongly regular graph with parameters ( v , k , t , λ, µ ) if and only if D satisfies the following conditions: - Every vertex has in-degree and out-degree k . - Every vertex x has t out-neighbors, all of which are also in-neighbors of x . 8 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Directed strongly regular graphs(DSRG) Definition A loopless directed graph D with v vertices is called directed strongly regular graph with parameters ( v , k , t , λ, µ ) if and only if D satisfies the following conditions: - Every vertex has in-degree and out-degree k . - Every vertex x has t out-neighbors, all of which are also in-neighbors of x . - The number of directed paths of length two from a vertex x to another vertex y is λ if there is an edge from x to y , and is µ if there is no edge from x to y . 9 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References DSRG-(8,3,2,1,1) 10 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Tactical Configuration Definition A tactical configuration is a triple T = ( P , B , I ) where - P is a v -element set, - B is a collection of k -element subsets of P (called ‘blocks’) with |B| = b , and - I = { ( p , B ) ∈ P × B : p ∈ B } such that each element of P (called a ‘point’) belongs to exactly r blocks. 11 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References The DSRGs from Degenerated Affine Planes l ( q ) denote the partial geometry obtained from AP ( q ) by Let AP considering all q 2 points and taking the lines of l parallel classes of the plane. 12 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References The DSRGs from Degenerated Affine Planes l ( q ) denote the partial geometry obtained from AP ( q ) by Let AP considering all q 2 points and taking the lines of l parallel classes of l ( q ) satisfies the following properties: the plane. Then AP 1 every point is incident with l lines, 2 every line is incident with q points, 3 any two points are incident with at most one line, 4 if p and L are non-incident point-line pair, there are exactly l − 1 lines containing p which meet L . l ( q ) is a pg ( q , l , l − 1). 5 AP 13 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Construction of DSRGs from Degenerated Affine Planes Theorem (1) l ( q )) be the directed graph with its vertex set Let D = D ( AP V ( D ) = { ( p , L ) ∈ P × L : p / ∈ L } , and directed edges given by ( p , L ) → ( p ′ , L ′ ) iff p ∈ L ′ . Then D is a directed strongly regular graph with parameters: ( lq 2 ( q − 1) , lq ( q − 1) , lq − l + 1 , ( l − 1)( q − 1) , lq − l + 1) . 14 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Affine Plane of Order 2 15 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References 16 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Antiflags 17 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Antiflags - A = ( L 1 , p 2 ) - B = ( L 1 , p 4 ) - C = ( L 2 , p 1 ) - D = ( L 2 , p 3 ) - E = ( L 3 , p 3 ) - F = ( L 3 , p 4 ) - G = ( L 4 , p 1 ) - H = ( L 4 , p 2 ) 18 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References DSRG(8,4,3,1,3) 19 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References New Graphs From Partial Geometries The new graphs given by these constructions have parameters (36 , 12 , 5 , 2 , 5) , (54 , 18 , 7 , 4 , 7) , (72 , 24 , 10 , 4 , 10) , (96 , 24 , 7 , 3 , 7) , (108 , 36 , 14 , 8 , 14) , (108 , 36 , 15 , 6 , 15) listed as feasible parameters with v ≤ 110 on “Parameters of directed strongly regular graphs” by S. Hobart and A. E. Brouwer at http : // homepages . cwi . nl / ∼ aeb / math / dsrg / dsrg . html . 20 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References An Interesting Example Arising From Projective Planes Let P be the point set of this projective plane. - For a point p ∈ P , let L p 0 , L p 1 , . . . , L pn denote the n + 1 lines passing through p . - Set B pi = L pi − { p } for i = 0 , 1 , . . . , n , - Then with B = { B pi : p ∈ P , i ∈ { 0 , 1 , . . . , n }} , the pair ( P , B ) forms a tactical configuration with parameters: ( v , b , k , r ) = ( n 2 + n + 1 , ( n + 1)( n 2 + n + 1) , n , n ( n + 1)) . 21 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References Construction of DSRG by Using Projective Planes Theorem (2) Let D be the directed graph with its vertex set V = { ( p , B pi ) ∈ P × B : p ∈ P , i ∈ { 0 , 1 , . . . , n } and adjacency defined by ( p , B pi ) → ( q , B qj ) if and only if p ∈ B qj . Then D is a directed strongly regular with the parameters ( v , k , t , λ, µ ) = (( n + 1)( n 2 + n + 1) , n ( n + 1) , n , n − 1 , n ) 22 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References DSRG-(21,6,2,1,2) Obtained from Fano Plane 23 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References DSRG-(21,6,2,1,2) Obtained from Fano Plane 1 23 , 45 , 67 2 13 , 46 , 57 3 12 , 56 , 47 4 15 , 26 , 37 5 14 , 36 , 27 6 17 , 35 , 24 7 16 , 25 , 34 24 / 37

Directed strongly regular graphs(DSRG) Combinatorial Designs Construction of Directed Strongly Regular Graphs References A General Idea 1 Consider the ( ls + 1)-element set P = { 1 , 2 , . . . , ls + 1 } . 2 For each i ∈ P , let B i = { B i 1 , B i 2 , . . . , B is } be a partition of P \ { i } into s parts (blocks) of equal size l . 3 Let B = � ls +1 i =1 B i = { B ig : 1 ≤ g ≤ s , 1 ≤ i ≤ ls + 1 } . Then the pair ( P , B ) forms a tactical configuration with parameters ( v , b , k , r ) = ( ls + 1 , s ( ls + 1) , l , ls ). 25 / 37