Some Families of Directed Strongly Regular Graphs Obtained from - - PowerPoint PPT Presentation

▶
some families of directed strongly regular graphs
SMART_READER_LITE
LIVE PREVIEW

Some Families of Directed Strongly Regular Graphs Obtained from - - PowerPoint PPT Presentation

Dec 28, 2023 263 likes •651 views

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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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

Overview

1 Directed Strongly Regular Graphs 2 / 37

slide-3
SLIDE 3

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

slide-4
SLIDE 4

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

slide-5
SLIDE 5

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

slide-6
SLIDE 6

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

slide-7
SLIDE 7

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

slide-8
SLIDE 8

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

slide-9
SLIDE 9

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

slide-10
SLIDE 10

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

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

10 / 37

slide-11
SLIDE 11

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

slide-12
SLIDE 12

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

The DSRGs from Degenerated Affine Planes

Let AP

l(q) denote the partial geometry obtained from AP(q) by

considering all q2 points and taking the lines of l parallel classes of the plane.

12 / 37

slide-13
SLIDE 13

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

The DSRGs from Degenerated Affine Planes

Let AP

l(q) denote the partial geometry obtained from AP(q) by

considering all q2 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).

13 / 37

slide-14
SLIDE 14

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

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: (lq2(q − 1), lq(q − 1), lq − l + 1, (l − 1)(q − 1), lq − l + 1).

14 / 37

slide-15
SLIDE 15

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

Affine Plane of Order 2

15 / 37

slide-16
SLIDE 16

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

slide-17
SLIDE 17

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

Antiflags

17 / 37

slide-18
SLIDE 18

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

Antiflags

  • A = (L1, p2)
  • B = (L1, p4)
  • C = (L2, p1)
  • D = (L2, p3)
  • E = (L3, p3)
  • F = (L3, p4)
  • G = (L4, p1)
  • H = (L4, p2)

18 / 37

slide-19
SLIDE 19

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

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

19 / 37

slide-20
SLIDE 20

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

slide-21
SLIDE 21

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 Lp0, Lp1, . . . , Lpn denote the n + 1 lines

passing through p.

  • Set Bpi = Lpi − {p} for i = 0, 1, . . . , n,
  • Then with B = {Bpi : p ∈ P, i ∈ {0, 1, . . . , n}}, the pair

(P, B) forms a tactical configuration with parameters: (v, b, k, r) = (n2 + n + 1, (n + 1)(n2 + n + 1), n, n(n + 1)).

21 / 37

slide-22
SLIDE 22

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, Bpi) ∈ P × B : p ∈ P, i ∈ {0, 1, . . . , n} and adjacency defined by (p, Bpi) → (q, Bqj) if and only if p ∈ Bqj. Then D is a directed strongly regular with the parameters (v, k, t, λ, µ) = ((n + 1)(n2 + n + 1), n(n + 1), n, n − 1, n)

22 / 37

slide-23
SLIDE 23

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

slide-24
SLIDE 24

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

slide-25
SLIDE 25

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

Bi = {Bi1, Bi2, . . . , Bis} be a partition of P \ {i} into s parts (blocks) of equal size l.

3 Let B = ls+1

i=1 Bi = {Big : 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

slide-26
SLIDE 26

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

Bi = {Bi1, Bi2, . . . , Bis} be a partition of P \ {i} into s parts (blocks) of equal size l.

3 Let B = ls+1

i=1 Bi = {Big : 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, 45 2 13, 45 3 12, 45 4 12, 35 5 12, 34

26 / 37

slide-27
SLIDE 27

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

Theorem (3) Let D = D(T ) be the directed graph with its vertex set V = {(g, B) : B ∈ Bg, g ∈ P} and adjacency defined by (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) = (ls2 + s, ls, l, l − 1, l).

27 / 37

slide-28
SLIDE 28

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

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

28 / 37

slide-29
SLIDE 29

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

DSRG-(10, 4, 2, 1, 2)

Let l = 2 and s = 2.

  • F be the set of tactical configurations.
  • Consider the action of S5 on F
  • T σ

1 = T2 if and only if V (T1)σ = V (T2) where

V (T )σ = {(iσ, (Bij)σ) : i ∈ P, Bij ∈ B} with natural action on Bij.

29 / 37

slide-30
SLIDE 30

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

The Block Sets of the Representatives of Seven Orbits

i B(T1) B(T2) B(T3) B(T4) B(T5) B(T6) B(T7) 1 23, 45 23, 45 23, 45 23, 45 23, 45 23, 45 23, 45 2 13, 45 13, 45 13, 45 14, 35 13, 45 13, 45 13, 45 3 12, 45 14, 25 12, 45 15, 24 14, 25 14, 25 14, 25 4 12, 35 12, 35 12, 35 13, 25 12, 35 12, 35 13, 25 5 12, 34 12, 34 13, 24 12, 34 14, 23 13, 24 14, 23

30 / 37

slide-31
SLIDE 31

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

Stabilizers and the Size of Orbits for the Action of S5 on F

T1 T2 T3 T4 T5 T6 T7 D8 C2 × C2 C2 C5 ⋊ C4 C2 C2 D10 15 30 60 6 60 60 12

31 / 37

slide-32
SLIDE 32

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

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

32 / 37

slide-33
SLIDE 33

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

[1] Bose, R. C., Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math. 13 (1963) 389–419. [2] Brouwer, A. E., Cohen, A. M. and Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989. [3] Brouwer, A. E. and Hobart, S., Directed strongly regular graph, in: Colburn, C. and Dinitz, J. (Eds.), Handbook of Combinatorial Designs, CRC Inc., Boca Raton, 868–875. [4] Duval, A., A directed graph version of strongly regular graphs, Journal of Combinatorial Theory, (A) 47 (1988), 71–100. [5] Duval, A. and Iourinski, D., Semidirect product constructions

  • f directed strongly regular graphs, Journal of Combinatorial

Theory, (A) 104 (2003) 157-167.

33 / 37

slide-34
SLIDE 34

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

[6] Fiedler, F, Klin, M. and Muzychuk, M., Small vertex-transitive directed strongly regular graphs, Discrete Mathematics 255 (2002) 87-115. [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. 69–87. [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, 1980–1993. [9] Hobart, S. and Brouwer, A. E., “Parameters of directed strongly regular graphs,” kept in A. Brouwer’s URL: [http://homepages.cwi.nl/∼aeb/math/dsrg/dsrg.html]

34 / 37

slide-35
SLIDE 35

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

[10] Hobart, S. and Shaw, T., A note on a family of directed strongly regular graphs, European Journal of Combinatorics 20 (1999), 819–820. [11] Jørgensen, L., Search for directed strongly regular graphs. Report R-99-2016 Department of Mathematical Sciences, Aalborg University, 1999. [12] Jørgensen, L., Directed strongly regular graphs with µ = λ, Discrete Mathematics, 231 (2001), no. 1–3, 289–293. [13] Jørgensen, L., Non-existence of directed strongly regular graphs, Discrete Mathematics, 264 (2003), 111–126. [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), 83–109.

35 / 37

slide-36
SLIDE 36

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

[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-10-1998, Technische Universitat Dresden. [16] Olmez, O. and Song, S. Y., Construction of directed strongly regular graphs, preprint. (arXiv:1006.5395v2 [math.CO].) [17] Olmez, O. and Song, S. Y., Some families of directed strongly regular graphs obtained from certain finite incidence structures, preprint. (arXiv:1102.1491[math.CO].)

36 / 37

slide-37
SLIDE 37

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

Thanks!

THANK YOU FOR YOUR ATTENTION Contact Sung Song: sysong@iastate.edu Contact me: oolmez@iastate.edu

37 / 37