Some Families of Directed Strongly Regular Graphs Obtained from Certain Finite Incidence Structures

Transcription

1 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

2 Overview 1 Directed Strongly Regular Graphs

3 Overview 1 Directed Strongly Regular Graphs 2

4 Overview 1 Directed Strongly Regular Graphs 2 3 Directed Strongly Regular Graphs Obtained from Affine Planes

5 Overview 1 Directed Strongly Regular Graphs 2 3 Directed Strongly Regular Graphs Obtained from Affine Planes 4 Directed Strongly Regular Graphs Obtained from Tactical Configurations

6 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:

7 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.

8 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.

9 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.

10 DSRG-(8,3,2,1,1)

11 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.

12 The DSRGs from Degenerated Affine Planes Let AP l (q) denote the partial geometry obtained from AP(q) by considering all q 2 points and taking the lines of l parallel classes of the plane.

13 The DSRGs from Degenerated Affine Planes Let AP l (q) denote the partial geometry obtained from AP(q) by considering all q 2 points and taking the lines of l parallel classes of the plane. Then AP l (q) satisfies the following properties: 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. 5 AP l (q) is a pg(q, l, l 1).

14 Construction of DSRGs from Degenerated Affine Planes Theorem (1) Let D = D(AP l (q)) be the directed graph with its vertex set 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).

15 Affine Plane of Order 2

16

17 Antiflags

18 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 )

19 DSRG(8,4,3,1,3)

20 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.

21 An Interesting Example Arising From Projective Planes Let P be the point set of this projective plane. - For a point p P, let L p0, L p1,..., 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)).

22 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)

23 DSRG-(21,6,2,1,2) Obtained from Fano Plane

24 DSRG-(21,6,2,1,2) Obtained from Fano Plane 1 23, 45, , 46, , 56, , 26, , 36, , 35, , 25, 34

25 A General Idea 1 Consider the (ls + 1)-element set P = {1, 2,..., ls + 1}. 2 For each i P, let B i = {B i1, B i2,..., 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).

26 A General Idea 1 Consider the (ls + 1)-element set P = {1, 2,..., ls + 1}. 2 For each i P, let B i = {B i1, B i2,..., 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). 1 23, , , , , 34

27 Theorem (3) Let D = D(T ) be the directed graph with its vertex set and adjacency defined by V = {(g, B) : B B g, g P} (g, B) (g, B ) if and only if g B. Then D is a directed strongly regular graph with the parameters: (v, k, t, λ, µ) = (ls 2 + s, ls, l, l 1, l).

28 Non-Isomorphic two DSRG-(10, 4, 2, 1, 2)

29 DSRG-(10, 4, 2, 1, 2) Let l = 2 and s = 2. - F be the set of tactical configurations. - Consider the action of S 5 on F - T σ 1 = T 2 if and only if V (T 1 ) σ = V (T 2 ) where V (T ) σ = {(i σ, (B ij ) σ ) : i P, B ij B} with natural action on B ij.

30 The Block Sets of the Representatives of Seven Orbits i B(T 1 ) B(T 2 ) B(T 3 ) B(T 4 ) B(T 5 ) B(T 6 ) B(T 7 ) 1 23, 45 23, 45 23, 45 23, 45 23, 45 23, 45 23, , 45 13, 45 13, 45 14, 35 13, 45 13, 45 13, , 45 14, 25 12, 45 15, 24 14, 25 14, 25 14, , 35 12, 35 12, 35 13, 25 12, 35 12, 35 13, , 34 12, 34 13, 24 12, 34 14, 23 13, 24 14, 23

31 Stabilizers and the Size of Orbits for the Action of S 5 on F T 1 T 2 T 3 T 4 T 5 T 6 T 7 D 8 C 2 C 2 C 2 C 5 C 4 C 2 C 2 D

32 New Graphs from Tactical Configuration The new graphs obtained from tactical configurations by similar methods have parameters: (45, 30, 22, 19, 22), (54, 36, 26, 23, 26), (72, 48, 34, 31, 34), (75, 60, 52, 47, 52), (81, 54, 38, 35, 38), (90, 30, 11, 8, 11), (90, 60, 44, 38, 44), (99, 66, 46, 43, 46), (100, 40, 18, 13, 18), (108, 36, 13, 10, 13), (108, 72, 50, 47, 50), (108, 72, 52, 46, 52), (108, 90, 80, 74, 80) 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

33 [1] Bose, R. C., Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math. 13 (1963) [2] Brouwer, A. E., Cohen, A. M. and Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, [3] Brouwer, A. E. and Hobart, S., Directed strongly regular graph, in: Colburn, C. and Dinitz, J. (Eds.), Handbook of, CRC Inc., Boca Raton, [4] Duval, A., A directed graph version of strongly regular graphs, Journal of Combinatorial Theory, (A) 47 (1988), [5] Duval, A. and Iourinski, D., Semidirect product constructions of directed strongly regular graphs, Journal of Combinatorial Theory, (A) 104 (2003)

34 [6] Fiedler, F, Klin, M. and Muzychuk, M., Small vertex-transitive directed strongly regular graphs, Discrete Mathematics 255 (2002) [7] Fiedler, F., Klin, M. and Pech, Ch., Directed strongly regular graphs as elements of coherent algebras, in: Denecke, K. and Vogel, H.-J. (Eds.), General Algebra and Discrete Mathematics, Shaker Verlag, Aachen, 1999, pp [8] Godsil, C. D., Hobart, S. A. and Martin, W. J., Representations of directed strongly regular graphs, European Journal of Combinatorics 28 (2007), no. 7, [9] Hobart, S. and Brouwer, A. E., Parameters of directed strongly regular graphs, kept in A. Brouwer s URL: [ aeb/math/dsrg/dsrg.html]

35 [10] Hobart, S. and Shaw, T., A note on a family of directed strongly regular graphs, European Journal of Combinatorics 20 (1999), [11] Jørgensen, L., Search for directed strongly regular graphs. Report R Department of Mathematical Sciences, Aalborg University, [12] Jørgensen, L., Directed strongly regular graphs with µ = λ, Discrete Mathematics, 231 (2001), no. 1 3, [13] Jørgensen, L., Non-existence of directed strongly regular graphs, Discrete Mathematics, 264 (2003), [14] Klin, M., Munemasa, A., Muzychuk, M. and Zieschang, P.-H., Directed strongly regular graphs obtained from coherent algebras, Linear Algebra and Its Applications 377 (2004),

36 [15] Klin, M., Pech, Ch. and Zieschang, P.-H., Flag algebras of block designs: I. Initial notions, Steiner 2-designs, and generalized quadrangles, preprint MATH-AL , Technische Universitat Dresden. [16] Olmez, O. and Song, S. Y., Construction of directed strongly regular graphs, preprint. (arxiv: v2 [math.co].) [17] Olmez, O. and Song, S. Y., Some families of directed strongly regular graphs obtained from certain finite incidence structures, preprint. (arxiv: [math.co].)

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