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Edge-regular graphs and regular cliques Gary Greaves Nanyang Technological University, Singapore 23rd May 2018 joint work with J. H. Koolen Gary Greaves Edge-regular graphs and regular cliques 1/13 Gary Greaves Edge-regular graphs and

  1. Edge-regular graphs and regular cliques Gary Greaves Nanyang Technological University, Singapore 23rd May 2018 joint work with J. H. Koolen Gary Greaves — Edge-regular graphs and regular cliques 1/13

  2. Gary Greaves — Edge-regular graphs and regular cliques 2/13

  3. k Gary Greaves — Edge-regular graphs and regular cliques 2/13

  4. k Gary Greaves — Edge-regular graphs and regular cliques 2/13

  5. k Gary Greaves — Edge-regular graphs and regular cliques 2/13

  6. k Gary Greaves — Edge-regular graphs and regular cliques 2/13

  7. k Gary Greaves — Edge-regular graphs and regular cliques 2/13

  8. k k = 6 Gary Greaves — Edge-regular graphs and regular cliques 2/13

  9. k k = 6 λ Gary Greaves — Edge-regular graphs and regular cliques 2/13

  10. k k = 6 λ Gary Greaves — Edge-regular graphs and regular cliques 2/13

  11. k k = 6 λ Gary Greaves — Edge-regular graphs and regular cliques 2/13

  12. k k = 6 λ Gary Greaves — Edge-regular graphs and regular cliques 2/13

  13. k k = 6 λ Gary Greaves — Edge-regular graphs and regular cliques 2/13

  14. k k = 6 λ λ = 3 Gary Greaves — Edge-regular graphs and regular cliques 2/13

  15. k k = 6 λ λ = 3 edge-regular erg ( 10, 6, 3 ) Gary Greaves — Edge-regular graphs and regular cliques 2/13

  16. 6 3 edge-regular erg ( 10, 6, 3 ) Gary Greaves — Edge-regular graphs and regular cliques 3/13

  17. 6 3 edge-regular erg ( 10, 6, 3 ) clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  18. 6 3 edge-regular erg ( 10, 6, 3 ) clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  19. 6 3 edge-regular erg ( 10, 6, 3 ) clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  20. 6 3 edge-regular erg ( 10, 6, 3 ) clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  21. 6 3 edge-regular erg ( 10, 6, 3 ) clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  22. 6 3 edge-regular erg ( 10, 6, 3 ) clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  23. 6 3 edge-regular erg ( 10, 6, 3 ) clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  24. 6 3 edge-regular erg ( 10, 6, 3 ) 2 - regular clique of order 4 Gary Greaves — Edge-regular graphs and regular cliques 3/13

  25. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. Gary Greaves — Edge-regular graphs and regular cliques 4/13

  26. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. Gary Greaves — Edge-regular graphs and regular cliques 4/13

  27. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 Gary Greaves — Edge-regular graphs and regular cliques 4/13

  28. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ Gary Greaves — Edge-regular graphs and regular cliques 4/13

  29. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ Gary Greaves — Edge-regular graphs and regular cliques 4/13

  30. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ Gary Greaves — Edge-regular graphs and regular cliques 4/13

  31. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ Gary Greaves — Edge-regular graphs and regular cliques 4/13

  32. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ Gary Greaves — Edge-regular graphs and regular cliques 4/13

  33. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ µ = 4 Gary Greaves — Edge-regular graphs and regular cliques 4/13

  34. Theorem (Neumaier 1981) Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ µ = 4 strongly regular srg ( 10, 6, 3, 4 ) Gary Greaves — Edge-regular graphs and regular cliques 4/13

  35. Question (Neumaier 1981) Is every edge-regular graph with a regular clique strongly regular? Gary Greaves — Edge-regular graphs and regular cliques 5/13

  36. Question (Neumaier 1981) Is every edge-regular graph with a regular clique strongly regular? Answer (GG and Koolen 2018) No. Gary Greaves — Edge-regular graphs and regular cliques 5/13

  37. Question (Neumaier 1981) Is every edge-regular graph with a regular clique strongly regular? Answer (GG and Koolen 2018) No. There exist infinitely many non-strongly-regular, edge-regular vertex-transitive graphs with regular cliques. Gary Greaves — Edge-regular graphs and regular cliques 5/13

  38. An example Gary Greaves — Edge-regular graphs and regular cliques 6/13

  39. Cayley graphs ◮ Let G be an (additive) group and S ⊆ G a (symmetric) generating subset, i.e., s ∈ S = ⇒ − s ∈ S and G = � S � . ◮ The Cayley graph Cay ( G , S ) has vertex set G and edge set {{ g , g + s } : g ∈ G and s ∈ S } . Example Γ = Cay ( Z 5 , S ) Generating set S = {− 1, 1 } 1 0 2 − 1 − 2 Gary Greaves — Edge-regular graphs and regular cliques 7/13

  40. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ( 11, 0 ) ( 11, ± 3 ) Gary Greaves — Edge-regular graphs and regular cliques 8/13

  41. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) : ( 11, 0 ) ( 11, ± 3 ) Gary Greaves — Edge-regular graphs and regular cliques 8/13

  42. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) : ( 11, 0 ) ( 11, ± 3 ) ( 00, 0 ) Gary Greaves — Edge-regular graphs and regular cliques 8/13

  43. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) : ( 11, 0 ) ( 11, ± 3 ) ( 10, 0 ) ( 01, 0 ) ( 11, 0 ) ( 00, 0 ) ( 11, 3 ) ( 01, 1 ) ( 10, 2 ) ( 10, − 2 ) ( 01, − 1 ) ( 11, − 3 ) Gary Greaves — Edge-regular graphs and regular cliques 8/13

  44. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) : ( 11, 0 ) ( 11, ± 3 ) ( 10, 0 ) ( 01, 0 ) ( 11, 0 ) ( 00, 0 ) ( 11, 3 ) ( 01, 1 ) ( 10, 2 ) ( 10, − 2 ) ( 01, − 1 ) ( 11, − 3 ) Gary Greaves — Edge-regular graphs and regular cliques 8/13

  45. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) ; ( 11, 0 ) ( 11, ± 3 ) ◮ Γ has a 1 -regular 4 -clique : Gary Greaves — Edge-regular graphs and regular cliques 9/13

  46. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) ; ( 11, 0 ) ( 11, ± 3 ) ◮ Γ has a 1 -regular 4 -clique : ( 01, 0 ) ( 00, 0 ) ( 10, 0 ) ( 11, 0 ) Gary Greaves — Edge-regular graphs and regular cliques 9/13

  47. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) ; ( 11, 0 ) ( 11, ± 3 ) ◮ Γ has a 1 -regular 4 -clique : ( 01, 0 ) ( 00, 0 ) ( 10, 0 ) ( a , b ) b � = 0 ( 11, 0 ) Gary Greaves — Edge-regular graphs and regular cliques 9/13

  48. An example Generating set S ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 01, 0 ) ( 01, ± 1 ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) ; ( 11, 0 ) ( 11, ± 3 ) ◮ Γ has a 1 -regular 4 -clique : ( 01, 0 ) ( ∗ , − b ) ( 00, 0 ) ( 10, 0 ) ( a , b ) b � = 0 ( 11, 0 ) Gary Greaves — Edge-regular graphs and regular cliques 9/13

  49. An example Generating set S ( 01, 0 ) ( 01, ± 1 ) ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) ; ( 11, 0 ) ( 11, ± 3 ) ◮ Γ has a 1 -regular 4 -clique; ◮ Γ is not strongly regular : Gary Greaves — Edge-regular graphs and regular cliques 10/13

  50. An example Generating set S ( 01, 0 ) ( 01, ± 1 ) ◮ Γ = Cay ( Z 2 2 ⊕ Z 7 , S ) ( 10, 0 ) ( 10, ± 2 ) ◮ Γ is edge-regular ( 28, 9, 2 ) ; ( 11, 0 ) ( 11, ± 3 ) ◮ Γ has a 1 -regular 4 -clique; ◮ Γ is not strongly regular : ( 11, 0 ) ( 01, 0 ) ( 01, 1 ) ( 11, 3 ) ( 00, 0 ) ( 10, 0 ) ( 10, − 2 ) ( 01, − 1 ) ( 11, − 3 ) ( 10, 2 ) Gary Greaves — Edge-regular graphs and regular cliques 10/13

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