Strongly Regular Cayley Graphs in Rank Two Abelian Groups Ken W. - PowerPoint PPT Presentation

The first feasibility condition for strongly regular graphs Fix a vertex x and count ordered pairs ( y, z ) such that y ∼ z and x ∼ y but x �∼ z , so that the vertices y and z induce the subgraph below. z y x k ( k − λ − 1) = ( v − k − 1) µ . (1) This is our first feasibility condition for the strongly regular graph parameters. Smith Cayley SRGs 2015 9 / 61

The first feasibility condition for strongly regular graphs Fix a vertex x and count ordered pairs ( y, z ) such that y ∼ z and x ∼ y but x �∼ z , so that the vertices y and z induce the subgraph below. z y x k ( k − λ − 1) = ( v − k − 1) µ . (1) This is our first feasibility condition for the strongly regular graph parameters. Smith Cayley SRGs 2015 9 / 61

The eigenvalues of a strongly regular graph Strongly regular graphs have a particularly interesting spectrum! If A is the adjacency matrix of a ( v, k, λ, µ ) strongly regular graph then A 2 = kI + λA + µ ( J − I − A ) . Rewrite this: A 2 − ( λ − µ ) A − ( k − µ ) I = µJ. If � v is an eigenvector with eigenvalue θ different from k then θ 2 − ( λ − µ ) θ − ( k − µ ) = 0 . Smith Cayley SRGs 2015 10 / 61

The eigenvalues of a strongly regular graph Strongly regular graphs have a particularly interesting spectrum! If A is the adjacency matrix of a ( v, k, λ, µ ) strongly regular graph then A 2 = kI + λA + µ ( J − I − A ) . Rewrite this: A 2 − ( λ − µ ) A − ( k − µ ) I = µJ. If � v is an eigenvector with eigenvalue θ different from k then θ 2 − ( λ − µ ) θ − ( k − µ ) = 0 . Smith Cayley SRGs 2015 10 / 61

The eigenvalues of a strongly regular graph Strongly regular graphs have a particularly interesting spectrum! If A is the adjacency matrix of a ( v, k, λ, µ ) strongly regular graph then A 2 = kI + λA + µ ( J − I − A ) . Rewrite this: A 2 − ( λ − µ ) A − ( k − µ ) I = µJ. If � v is an eigenvector with eigenvalue θ different from k then θ 2 − ( λ − µ ) θ − ( k − µ ) = 0 . Smith Cayley SRGs 2015 10 / 61

The eigenvalues of a strongly regular graph If � v is an eigenvector with eigenvalue θ different from k then θ 2 − ( λ − µ ) θ − ( k − µ ) I = 0 . 1 (9 , 4 , 1 , 2) has eigenvalue k = 4 and the others are roots of θ 2 + θ − 2 = 0 . So θ = 1 , − 2 2 (16 , 5 , 0 , 2) has eigenvalue k = 5 and the others are roots of θ 2 + 2 θ − 3 = 0 . So θ = 1 , − 3 Smith Cayley SRGs 2015 11 / 61

The eigenvalues of a strongly regular graph If � v is an eigenvector with eigenvalue θ different from k then θ 2 − ( λ − µ ) θ − ( k − µ ) I = 0 . 1 (9 , 4 , 1 , 2) has eigenvalue k = 4 and the others are roots of θ 2 + θ − 2 = 0 . So θ = 1 , − 2 2 (16 , 5 , 0 , 2) has eigenvalue k = 5 and the others are roots of θ 2 + 2 θ − 3 = 0 . So θ = 1 , − 3 Smith Cayley SRGs 2015 11 / 61

The eigenvalues of a strongly regular graph If � v is an eigenvector with eigenvalue θ different from k then θ 2 − ( λ − µ ) θ − ( k − µ ) I = 0 . 1 (9 , 4 , 1 , 2) has eigenvalue k = 4 and the others are roots of θ 2 + θ − 2 = 0 . So θ = 1 , − 2 2 (16 , 5 , 0 , 2) has eigenvalue k = 5 and the others are roots of θ 2 + 2 θ − 3 = 0 . So θ = 1 , − 3 Smith Cayley SRGs 2015 11 / 61

The eigenvalues of the (16 , 6 , 2 , 2) Lattice Graph 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 Smith Cayley SRGs 2015 12 / 61 0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0

The eigenvalues of the (16 , 6 , 2 , 2) Lattice Graph   J − I I I I I J − I I I   A =   I I J − I I   I I I J − I has eigenvalues k = 6 and the roots of x 2 − 4 : 6 , 2 , 2 , 2 , 2 , 2 , 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 . Smith Cayley SRGs 2015 13 / 61

The eigenvalues of the (16 , 6 , 2 , 2) Lattice Graph   J − I I I I I J − I I I   A =   I I J − I I   I I I J − I has eigenvalues k = 6 and the roots of x 2 − 4 : 6 , 2 , 2 , 2 , 2 , 2 , 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 , − 2 . Smith Cayley SRGs 2015 13 / 61

The eigenvalues of the (16 , 6 , 2 , 2) Lattice Graph   J − I I I I I J − I I I   A =   I I J − I I   I I I J − I has eigenvalues k = 6 and the roots of x 2 − 4 : 6 , 2 6 , − 2 9 . Smith Cayley SRGs 2015 13 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 14 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 15 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 16 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 17 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 18 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 19 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 20 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph G = Z 4 ⊕ Z 4 , S = { 10 , 20 , 30 , 01 , 02 , 03 } 00 10 20 30 01 11 21 31 02 12 22 32 03 13 23 33 Smith Cayley SRGs 2015 21 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph G = C 4 × C 4 , S = x + x 2 + x 3 + y + y 2 + y 3 x 1 x 2 x 3 y xy x 2 y x 3 y y 2 xy 2 x 2 y 2 x 3 y 2 y 3 xy 3 x 2 y 3 x 3 y 3 Smith Cayley SRGs 2015 22 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph G = C 4 × C 4 , S = x + x 2 + x 3 + y + y 2 + y 3 x 1 x 2 x 3 y xy x 2 y x 3 y y 2 xy 2 x 2 y 2 x 3 y 2 y 3 xy 3 x 2 y 3 x 3 y 3 Smith Cayley SRGs 2015 23 / 61

The (16 , 6 , 2 , 2) Shrikhande Graph as a Cayley Graph G = C 4 × C 4 , S = x + x 3 + y + y 3 + xy + x 3 y 3 x 1 x 2 x 3 y xy x 2 y x 3 y y 2 xy 2 x 2 y 2 x 3 y 2 y 3 xy 3 x 2 y 3 x 3 y 3 Smith Cayley SRGs 2015 24 / 61

The (16 , 6 , 2 , 2) Shrikhande Graph as a Cayley Graph G = C 4 × C 4 , S = x + x 3 + y + y 3 + xy + x 3 y 3 x 1 x 2 x 3 y xy x 2 y x 3 y y 2 xy 2 x 2 y 2 x 3 y 2 y 3 xy 3 x 2 y 3 x 3 y 3 Smith Cayley SRGs 2015 25 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph Recall   J − I I I I I J − I I I   A =   I I J − I I   I I I J − I has eigenvalues 6 , 2 6 , − 2 9 . Smith Cayley SRGs 2015 26 / 61

The (16 , 6 , 2 , 2) Lattice Graph as a Cayley Graph Recall   J − I I I I I J − I I I   A =   I I J − I I   I I I J − I has eigenvalues 6 , 2 6 , − 2 9 . Smith Cayley SRGs 2015 26 / 61

The general problem We seek strongly regular Cayley graphs (partial difference sets) with m 2 vertices and degree k = r ( m − 1) or k = r ( m + 1) (for various values of r ) in abelian groups with exponent higher than that given by structures in the additive group of a field. In particular, we are interested in groups of order p 2 n with exponent higher than p . The case p = 2 seems to be distinct from the odd prime cases. Jim Davis (U. Richmond), John Polhill (Bloomsburg U.) and others have constructed PDS in direct products of cyclic groups of order 4. The highest that the exponent could be is 2 n . (Polhill & Davis) Can we construct PDS in the group G = C 2 n × C 2 n ? Smith Cayley SRGs 2015 27 / 61

The general problem We seek strongly regular Cayley graphs (partial difference sets) with m 2 vertices and degree k = r ( m − 1) or k = r ( m + 1) (for various values of r ) in abelian groups with exponent higher than that given by structures in the additive group of a field. In particular, we are interested in groups of order p 2 n with exponent higher than p . The case p = 2 seems to be distinct from the odd prime cases. Jim Davis (U. Richmond), John Polhill (Bloomsburg U.) and others have constructed PDS in direct products of cyclic groups of order 4. The highest that the exponent could be is 2 n . (Polhill & Davis) Can we construct PDS in the group G = C 2 n × C 2 n ? Smith Cayley SRGs 2015 27 / 61

The general problem We seek strongly regular Cayley graphs (partial difference sets) with m 2 vertices and degree k = r ( m − 1) or k = r ( m + 1) (for various values of r ) in abelian groups with exponent higher than that given by structures in the additive group of a field. In particular, we are interested in groups of order p 2 n with exponent higher than p . The case p = 2 seems to be distinct from the odd prime cases. Jim Davis (U. Richmond), John Polhill (Bloomsburg U.) and others have constructed PDS in direct products of cyclic groups of order 4. The highest that the exponent could be is 2 n . (Polhill & Davis) Can we construct PDS in the group G = C 2 n × C 2 n ? Smith Cayley SRGs 2015 27 / 61

The general problem We seek strongly regular Cayley graphs (partial difference sets) with m 2 vertices and degree k = r ( m − 1) or k = r ( m + 1) (for various values of r ) in abelian groups with exponent higher than that given by structures in the additive group of a field. In particular, we are interested in groups of order p 2 n with exponent higher than p . The case p = 2 seems to be distinct from the odd prime cases. Jim Davis (U. Richmond), John Polhill (Bloomsburg U.) and others have constructed PDS in direct products of cyclic groups of order 4. The highest that the exponent could be is 2 n . (Polhill & Davis) Can we construct PDS in the group G = C 2 n × C 2 n ? Smith Cayley SRGs 2015 27 / 61

The general problem We seek strongly regular Cayley graphs (partial difference sets) with m 2 vertices and degree k = r ( m − 1) or k = r ( m + 1) (for various values of r ) in abelian groups with exponent higher than that given by structures in the additive group of a field. In particular, we are interested in groups of order p 2 n with exponent higher than p . The case p = 2 seems to be distinct from the odd prime cases. Jim Davis (U. Richmond), John Polhill (Bloomsburg U.) and others have constructed PDS in direct products of cyclic groups of order 4. The highest that the exponent could be is 2 n . (Polhill & Davis) Can we construct PDS in the group G = C 2 n × C 2 n ? Smith Cayley SRGs 2015 27 / 61

The general problem We seek strongly regular Cayley graphs (partial difference sets) with m 2 vertices and degree k = r ( m − 1) or k = r ( m + 1) (for various values of r ) in abelian groups with exponent higher than that given by structures in the additive group of a field. In particular, we are interested in groups of order p 2 n with exponent higher than p . The case p = 2 seems to be distinct from the odd prime cases. Jim Davis (U. Richmond), John Polhill (Bloomsburg U.) and others have constructed PDS in direct products of cyclic groups of order 4. The highest that the exponent could be is 2 n . (Polhill & Davis) Can we construct PDS in the group G = C 2 n × C 2 n ? Smith Cayley SRGs 2015 27 / 61

The general problem We seek strongly regular Cayley graphs (partial difference sets) with m 2 vertices and degree k = r ( m − 1) or k = r ( m + 1) (for various values of r ) in abelian groups with exponent higher than that given by structures in the additive group of a field. In particular, we are interested in groups of order p 2 n with exponent higher than p . The case p = 2 seems to be distinct from the odd prime cases. Jim Davis (U. Richmond), John Polhill (Bloomsburg U.) and others have constructed PDS in direct products of cyclic groups of order 4. The highest that the exponent could be is 2 n . (Polhill & Davis) Can we construct PDS in the group G = C 2 n × C 2 n ? Smith Cayley SRGs 2015 27 / 61

The characters of C 4 × C 4 and the Lattice Graph The 16 elements of C 4 × C 4 can be mapped into C in 16 different ways! S = ( x + x 2 + x 3 ) + ( y + y 2 + y 3 ) χ x y χ 0 , 0 1 1 3 + 3 = 6 χ 1 , 0 i 1 − 1 + 3 = 2 χ 2 , 0 − 1 1 − 1 + 3 = 2 χ 3 , 0 − i 1 − 1 + 3 = 2 χ 0 , 1 1 i 3 − 1 = 2 χ 1 , 1 i i − 1 − 1 = − 2 χ 2 , 1 − 1 i − 1 − 1 = − 2 χ 3 , 1 − i i − 1 − 1 = − 2 χ 0 , 2 1 − 1 3 − 1 = 2 χ 1 , 2 i − 1 − 1 − 1 = − 2 χ 2 , 2 − 1 − 1 − 1 − 1 = − 2 χ 3 , 2 − i − 1 − 1 − 1 = − 2 χ 0 , 3 1 − i 3 − 1 = 2 χ 1 , 3 i − i − 1 − 1 = − 2 χ 2 , 3 − 1 − i − 1 − 1 = − 2 χ 3 , 3 − i − i − 1 − 1 = − 2 Smith Cayley SRGs 2015 28 / 61

The characters of C 4 × C 4 and the Lattice Graph The 16 elements of C 4 × C 4 can be mapped into C in 16 different ways! S = ( x + x 2 + x 3 ) + ( y + y 2 + y 3 ) χ x y χ 0 , 0 1 1 3 + 3 = 6 χ 1 , 0 i 1 − 1 + 3 = 2 χ 2 , 0 − 1 1 − 1 + 3 = 2 χ 3 , 0 − i 1 − 1 + 3 = 2 χ 0 , 1 1 i 3 − 1 = 2 χ 1 , 1 i i − 1 − 1 = − 2 χ 2 , 1 − 1 i − 1 − 1 = − 2 χ 3 , 1 − i i − 1 − 1 = − 2 χ 0 , 2 1 − 1 3 − 1 = 2 χ 1 , 2 i − 1 − 1 − 1 = − 2 χ 2 , 2 − 1 − 1 − 1 − 1 = − 2 χ 3 , 2 − i − 1 − 1 − 1 = − 2 χ 0 , 3 1 − i 3 − 1 = 2 χ 1 , 3 i − i − 1 − 1 = − 2 χ 2 , 3 − 1 − i − 1 − 1 = − 2 χ 3 , 3 − i − i − 1 − 1 = − 2 Smith Cayley SRGs 2015 28 / 61

The characters of C 4 × C 4 and the Lattice Graph Let’s focus on the characters which map S to 2 . S = ( x + x 2 + x 3 ) + ( y + y 2 + y 3 ) χ x y χ 0 , 0 1 1 3 + 3 = 6 χ 1 , 0 i 1 − 1 + 3 = 2 χ 2 , 0 − 1 1 − 1 + 3 = 2 χ 3 , 0 − i 1 − 1 + 3 = 2 χ 0 , 1 1 i 3 − 1 = 2 χ 1 , 1 i i − 1 − 1 = − 2 χ 2 , 1 − 1 i − 1 − 1 = − 2 χ 3 , 1 − i i − 1 − 1 = − 2 χ 0 , 2 1 − 1 3 − 1 = 2 χ 1 , 2 i − 1 − 1 − 1 = − 2 χ 2 , 2 − 1 − 1 − 1 − 1 = − 2 χ 3 , 2 − i − 1 − 1 − 1 = − 2 χ 0 , 3 1 − i 3 − 1 = 2 χ 1 , 3 i − i − 1 − 1 = − 2 χ 2 , 3 − 1 − i − 1 − 1 = − 2 χ 3 , 3 − i − i − 1 − 1 = − 2 Smith Cayley SRGs 2015 29 / 61

The dual Cayley graph S ∗ = ( χ 1 , 0 + χ 2 , 0 + χ 3 , 0 ) + ( χ 0 , 1 + χ 0 , 2 + χ 0 , 3 ) in G ∗ . S = ( x + x 2 + x 3 ) + ( y + y 2 + y 3 ) χ x y χ 0 , 0 1 1 3 + 3 = 6 χ 1 , 0 i 1 − 1 + 3 = 2 χ 2 , 0 − 1 1 − 1 + 3 = 2 χ 3 , 0 − i 1 − 1 + 3 = 2 χ 0 , 1 1 i 3 − 1 = 2 χ 1 , 1 i i − 1 − 1 = − 2 χ 2 , 1 − 1 i − 1 − 1 = − 2 χ 3 , 1 − i i − 1 − 1 = − 2 χ 0 , 2 1 − 1 3 − 1 = 2 χ 1 , 2 i − 1 − 1 − 1 = − 2 χ 2 , 2 − 1 − 1 − 1 − 1 = − 2 χ 3 , 2 − i − 1 − 1 − 1 = − 2 χ 0 , 3 1 − i 3 − 1 = 2 χ 1 , 3 i − i − 1 − 1 = − 2 χ 2 , 3 − 1 − i − 1 − 1 = − 2 χ 3 , 3 − i − i − 1 − 1 = − 2 Smith Cayley SRGs 2015 30 / 61

The characters of C 4 × C 4 and the Shrikhande Graph Here are the values of the characters of C 4 × C 4 on the Shrikhande Cayley Graph. χ x y S = ( x + x 3 ) + ( y + y 3 ) + ( xy + x 3 y 3 ) χ 0 , 0 1 1 2 + 2 + 2 = 6 χ 1 , 0 i 1 0 + 2 + 0 = 2 χ 2 , 0 − 1 1 − 2 + 2 − 2 = − 2 χ 3 , 0 − i 1 0 + 2 + 0 = 2 χ 0 , 1 1 i 2 + 0 + 0 = 2 χ 1 , 1 i i 0 + 0 − 2 = − 2 χ 2 , 1 − 1 i − 2 + 0 + 0 = − 2 χ 3 , 1 − i i 0 + 0 + 2 = 2 χ 0 , 2 1 − 1 2 − 2 − 2 = − 2 χ 1 , 2 i − 1 0 − 2 + 0 = − 2 χ 2 , 2 − 1 − 1 − 2 − 2 + 2 = − 2 χ 3 , 2 − i − 1 0 − 2 + 0 = − 2 χ 0 , 3 1 − i 2 + 0 + 0 = 2 χ 1 , 3 i − i 0 + 0 + 2 = 2 χ 2 , 3 − 1 − i − 2 + 0 + 0 = − 2 χ 3 , 3 − i − i 0 + 0 + − 2 = − 2 Smith Cayley SRGs 2015 31 / 61

The characters of C 4 × C 4 and the Shrikhande Graph Here are the values of the characters of C 4 × C 4 on the Shrikhande Cayley Graph. χ x y S = ( x + x 3 ) + ( y + y 3 ) + ( xy + x 3 y 3 ) χ 0 , 0 1 1 2 + 2 + 2 = 6 χ 1 , 0 i 1 0 + 2 + 0 = 2 χ 2 , 0 − 1 1 − 2 + 2 − 2 = − 2 χ 3 , 0 − i 1 0 + 2 + 0 = 2 χ 0 , 1 1 i 2 + 0 + 0 = 2 χ 1 , 1 i i 0 + 0 − 2 = − 2 χ 2 , 1 − 1 i − 2 + 0 + 0 = − 2 χ 3 , 1 − i i 0 + 0 + 2 = 2 χ 0 , 2 1 − 1 2 − 2 − 2 = − 2 χ 1 , 2 i − 1 0 − 2 + 0 = − 2 χ 2 , 2 − 1 − 1 − 2 − 2 + 2 = − 2 χ 3 , 2 − i − 1 0 − 2 + 0 = − 2 χ 0 , 3 1 − i 2 + 0 + 0 = 2 χ 1 , 3 i − i 0 + 0 + 2 = 2 χ 2 , 3 − 1 − i − 2 + 0 + 0 = − 2 χ 3 , 3 − i − i 0 + 0 + − 2 = − 2 Smith Cayley SRGs 2015 31 / 61

The dual of the Shrikhande Graph S ∗ = ( χ 1 , 0 + χ 3 , 0 ) + ( χ 0 , 1 + χ 0 , 3 ) + ( χ 1 , 1 + χ 3 , 3 ) χ x y S = ( x + x 3 ) + ( y + y 3 ) + ( xy + x 3 y 3 ) χ 0 , 0 1 1 2 + 2 + 2 = 6 χ 1 , 0 i 1 0 + 2 + 0 = 2 χ 2 , 0 − 1 1 − 2 + 2 − 2 = − 2 χ 3 , 0 − i 1 0 + 2 + 0 = 2 χ 0 , 1 1 i 2 + 0 + 0 = 2 χ 1 , 1 i i 0 + 0 − 2 = − 2 χ 2 , 1 − 1 i − 2 + 0 + 0 = − 2 χ 3 , 1 − i i 0 + 0 + 2 = 2 χ 0 , 2 1 − 1 2 − 2 − 2 = − 2 χ 1 , 2 i − 1 0 − 2 + 0 = − 2 χ 2 , 2 − 1 − 1 − 2 − 2 + 2 = − 2 χ 3 , 2 − i − 1 0 − 2 + 0 = − 2 χ 0 , 3 1 − i 2 + 0 + 0 = 2 χ 1 , 3 i − i 0 + 0 + 2 = 2 χ 2 , 3 − 1 − i − 2 + 0 + 0 = − 2 χ 3 , 3 − i − i 0 + 0 + − 2 = − 2 Smith Cayley SRGs 2015 32 / 61

The group Z 4 ⊕ Z 4 and its dual 00 20 22 02 10 12 11 13 01 21 30 32 31 31 03 23 Smith Cayley SRGs 2015 33 / 61

The rook graph 00 20 22 02 10 12 11 13 01 21 30 32 31 31 03 23 Smith Cayley SRGs 2015 34 / 61

The Shrikhande graph 00 20 22 02 10 12 11 13 01 21 30 32 31 31 03 23 Smith Cayley SRGs 2015 35 / 61

Rational idempotents and a particular directed graphs We define an equivalence relationship on ∼ on G by g ∼ h if there exists s ∈ Z , GCD ( s, | G | ) = 1 such that h = g s . The dual group G ∗ of characters of G is isomorphic to G . On G ∗ we define an equivalence relation by χ ∼ ψ if Ker ( χ ) = Ker ( ψ ) . It turns out that this is the same relationship as that described above ( h = g s ) Smith Cayley SRGs 2015 36 / 61

Rational idempotents and a particular directed graphs We define an equivalence relationship on ∼ on G by g ∼ h if there exists s ∈ Z , GCD ( s, | G | ) = 1 such that h = g s . The dual group G ∗ of characters of G is isomorphic to G . On G ∗ we define an equivalence relation by χ ∼ ψ if Ker ( χ ) = Ker ( ψ ) . It turns out that this is the same relationship as that described above ( h = g s ) Smith Cayley SRGs 2015 36 / 61

Rational idempotents and a particular directed graphs We define an equivalence relationship on ∼ on G by g ∼ h if there exists s ∈ Z , GCD ( s, | G | ) = 1 such that h = g s . The dual group G ∗ of characters of G is isomorphic to G . On G ∗ we define an equivalence relation by χ ∼ ψ if Ker ( χ ) = Ker ( ψ ) . It turns out that this is the same relationship as that described above ( h = g s ) Smith Cayley SRGs 2015 36 / 61

Rational idempotents and a particular directed graphs We define an equivalence relationship on ∼ on G by g ∼ h if there exists s ∈ Z , GCD ( s, | G | ) = 1 such that h = g s . The dual group G ∗ of characters of G is isomorphic to G . On G ∗ we define an equivalence relation by χ ∼ ψ if Ker ( χ ) = Ker ( ψ ) . It turns out that this is the same relationship as that described above ( h = g s ) Smith Cayley SRGs 2015 36 / 61

The rational idempotents graph For each character χ ∈ G ∗ , the rational idempotent corresponding to χ is defined as 1 � e χ := χ ( g ) g. (2) | G | g ∈ G However, we are interested in Cayley graphs for which the eigenvalues are integers and so we want to work with rational idempotents. Given a character χ ∈ G ∗ , defined the rational idempotent corresponding to χ as � [ e χ ] := e χ . (3) χ ′ ∼ χ Each element of G ∗ / ∼ provides us with a rational idempotent; the rational idempotents are in a natural one-to-one correspondence with G ∗ / ∼ . Smith Cayley SRGs 2015 37 / 61

The rational idempotents graph For each character χ ∈ G ∗ , the rational idempotent corresponding to χ is defined as 1 � e χ := χ ( g ) g. (2) | G | g ∈ G However, we are interested in Cayley graphs for which the eigenvalues are integers and so we want to work with rational idempotents. Given a character χ ∈ G ∗ , defined the rational idempotent corresponding to χ as � [ e χ ] := e χ . (3) χ ′ ∼ χ Each element of G ∗ / ∼ provides us with a rational idempotent; the rational idempotents are in a natural one-to-one correspondence with G ∗ / ∼ . Smith Cayley SRGs 2015 37 / 61

The rational idempotents graph For each character χ ∈ G ∗ , the rational idempotent corresponding to χ is defined as 1 � e χ := χ ( g ) g. (2) | G | g ∈ G However, we are interested in Cayley graphs for which the eigenvalues are integers and so we want to work with rational idempotents. Given a character χ ∈ G ∗ , defined the rational idempotent corresponding to χ as � [ e χ ] := e χ . (3) χ ′ ∼ χ Each element of G ∗ / ∼ provides us with a rational idempotent; the rational idempotents are in a natural one-to-one correspondence with G ∗ / ∼ . Smith Cayley SRGs 2015 37 / 61

The rational idempotents graph For each character χ ∈ G ∗ , the rational idempotent corresponding to χ is defined as 1 � e χ := χ ( g ) g. (2) | G | g ∈ G However, we are interested in Cayley graphs for which the eigenvalues are integers and so we want to work with rational idempotents. Given a character χ ∈ G ∗ , defined the rational idempotent corresponding to χ as � [ e χ ] := e χ . (3) χ ′ ∼ χ Each element of G ∗ / ∼ provides us with a rational idempotent; the rational idempotents are in a natural one-to-one correspondence with G ∗ / ∼ . Smith Cayley SRGs 2015 37 / 61

An example The rational idempotents of C 4 × C 4 may be represented as 4 × 4 arrays. For example, the idempotent corresponding to the trivial representation is   1 1 1 1 16 � x, y � = 1 1 1 1 1 1     1 1 1 1 16   1 1 1 1 while the character which maps x to − 1 and y to 1 has idempotent   1 − 1 1 − 1 16(1 − x ) � x 2 , y � = 1 1 1 − 1 1 − 1     1 − 1 1 − 1 16   1 − 1 1 − 1 Smith Cayley SRGs 2015 38 / 61

An example The rational idempotents of C 4 × C 4 may be represented as 4 × 4 arrays. For example, the idempotent corresponding to the trivial representation is   1 1 1 1 16 � x, y � = 1 1 1 1 1 1     1 1 1 1 16   1 1 1 1 while the character which maps x to − 1 and y to 1 has idempotent   1 − 1 1 − 1 16(1 − x ) � x 2 , y � = 1 1 1 − 1 1 − 1     1 − 1 1 − 1 16   1 − 1 1 − 1 Smith Cayley SRGs 2015 38 / 61

An example The rational idempotents of C 4 × C 4 may be represented as 4 × 4 arrays. For example, the idempotent corresponding to the trivial representation is   1 1 1 1 16 � x, y � = 1 1 1 1 1 1     1 1 1 1 16   1 1 1 1 while the character which maps x to − 1 and y to 1 has idempotent   1 − 1 1 − 1 16(1 − x ) � x 2 , y � = 1 1 1 − 1 1 − 1     1 − 1 1 − 1 16   1 − 1 1 − 1 Smith Cayley SRGs 2015 38 / 61

An example The character which maps x to i and y to i is equivalent to the character which maps both x and y to − i . This pair of characters give rational idempotent   2 0 − 2 0 1 8(1 − x ) � x 2 , y � = 1 0 − 2 0 2     − 2 0 2 0 16   0 2 0 − 2 Smith Cayley SRGs 2015 39 / 61

An example The character which maps x to i and y to i is equivalent to the character which maps both x and y to − i . This pair of characters give rational idempotent   2 0 − 2 0 1 8(1 − x ) � x 2 , y � = 1 0 − 2 0 2     − 2 0 2 0 16   0 2 0 − 2 Smith Cayley SRGs 2015 39 / 61

The rational idempotents graph Suppose S ∗ = { χ 1 , 0 , χ 3 , 0 , χ 1 , 1 , χ 3 , 3 , χ 0 , 1 χ 0 , 3 } , so that the characters in S ∗ map S to 2 while all the other nontrivial characters map S to − 2 . Then S = 6[ e 0 , 0 ] − 2( S ∗ ) + 2 S ∗ = − 2 + 8[ e 0 , 0 ] + 4([ e 1 , 0 ] + [ e 1 , 1 ] + [ e 0 , 1 ]) .       − 32 0 0 0 8 8 8 8 24 8 − 8 8 = 1 0 0 0 0 8 8 8 8 8 8 − 8 − 8       � �  +  +       0 0 0 0 8 8 8 8 − 8 − 8 − 8 − 8 16     0 0 0 0 8 8 8 8 8 8 − 8 − 8   0 1 0 1 1 1 0 0  = x + x 3 + y + y 3 + xy + x 3 y 3 .   =   0 0 0 0  1 0 0 1 Smith Cayley SRGs 2015 40 / 61

The rational idempotents graph Suppose S ∗ = { χ 1 , 0 , χ 3 , 0 , χ 1 , 1 , χ 3 , 3 , χ 0 , 1 χ 0 , 3 } , so that the characters in S ∗ map S to 2 while all the other nontrivial characters map S to − 2 . Then S = 6[ e 0 , 0 ] − 2( S ∗ ) + 2 S ∗ = − 2 + 8[ e 0 , 0 ] + 4([ e 1 , 0 ] + [ e 1 , 1 ] + [ e 0 , 1 ]) .       − 32 0 0 0 8 8 8 8 24 8 − 8 8 = 1 0 0 0 0 8 8 8 8 8 8 − 8 − 8       � �  +  +       0 0 0 0 8 8 8 8 − 8 − 8 − 8 − 8 16     0 0 0 0 8 8 8 8 8 8 − 8 − 8   0 1 0 1 1 1 0 0  = x + x 3 + y + y 3 + xy + x 3 y 3 .   =   0 0 0 0  1 0 0 1 Smith Cayley SRGs 2015 40 / 61

The rational idempotents graph Suppose S ∗ = { χ 1 , 0 , χ 3 , 0 , χ 1 , 1 , χ 3 , 3 , χ 0 , 1 χ 0 , 3 } , so that the characters in S ∗ map S to 2 while all the other nontrivial characters map S to − 2 . Then S = 6[ e 0 , 0 ] − 2( S ∗ ) + 2 S ∗ = − 2 + 8[ e 0 , 0 ] + 4([ e 1 , 0 ] + [ e 1 , 1 ] + [ e 0 , 1 ]) .       − 32 0 0 0 8 8 8 8 24 8 − 8 8 = 1 0 0 0 0 8 8 8 8 8 8 − 8 − 8       � �  +  +       0 0 0 0 8 8 8 8 − 8 − 8 − 8 − 8 16     0 0 0 0 8 8 8 8 8 8 − 8 − 8   0 1 0 1 1 1 0 0  = x + x 3 + y + y 3 + xy + x 3 y 3 .   =   0 0 0 0  1 0 0 1 Smith Cayley SRGs 2015 40 / 61

The rational idempotents graph Suppose S ∗ = { χ 1 , 0 , χ 3 , 0 , χ 1 , 1 , χ 3 , 3 , χ 0 , 1 χ 0 , 3 } , so that the characters in S ∗ map S to 2 while all the other nontrivial characters map S to − 2 . Then S = 6[ e 0 , 0 ] − 2( S ∗ ) + 2 S ∗ = − 2 + 8[ e 0 , 0 ] + 4([ e 1 , 0 ] + [ e 1 , 1 ] + [ e 0 , 1 ]) .       − 32 0 0 0 8 8 8 8 24 8 − 8 8 = 1 0 0 0 0 8 8 8 8 8 8 − 8 − 8       � �  +  +       0 0 0 0 8 8 8 8 − 8 − 8 − 8 − 8 16     0 0 0 0 8 8 8 8 8 8 − 8 − 8   0 1 0 1 1 1 0 0  = x + x 3 + y + y 3 + xy + x 3 y 3 .   =   0 0 0 0  1 0 0 1 Smith Cayley SRGs 2015 40 / 61

The Shrikhande graph ... 00 20 22 02 10 12 11 13 01 21 30 32 31 31 03 23 Smith Cayley SRGs 2015 41 / 61

... and its dual χ 00 χ 20 χ 22 χ 02 χ 10 χ 12 χ 11 χ 13 χ 01 χ 21 χ 30 χ 32 χ 31 χ 31 χ 03 χ 23 Smith Cayley SRGs 2015 42 / 61

The Clebsch graph and its dual 00 20 22 02 10 12 11 13 01 21 30 32 31 31 03 23 Smith Cayley SRGs 2015 43 / 61

... and its dual χ 00 χ 20 χ 22 χ 02 χ 10 χ 12 χ 11 χ 13 χ 01 χ 21 χ 30 χ 32 χ 31 χ 31 χ 03 χ 23 Smith Cayley SRGs 2015 44 / 61

The group Z 8 × Z 8 and its dual 00 40 44 04 20 24 22 26 02 42 10 14 12 16 11 15 13 17 01 41 21 61 Smith Cayley SRGs 2015 45 / 61

The unique (64 , 18 , 2 , 6) Cayley Graph in Z 8 × Z 8 S = { 10 , 30 , 50 , 70 }∪{ 11 , 33 , 55 , 77 }∪{ 01 , 03 , 05 , 07 }∪{ 24 , 64 }∪{ 26 , 62 }∪{ 42 , 46 } 24 26 42 10 11 01 Smith Cayley SRGs 2015 46 / 61

The group ring element of a rational idempotent 1 − 1 − 1 1 − 2 2 4 − 4 − 8 8 (16 by 16, order 16 characters) Smith Cayley SRGs 2015 47 / 61

G = C 16 × C 16 , k = 190 , θ i = 4 , − 12 Smith Cayley SRGs 2015 48 / 61

Extending this to larger groups Strongly regular Cayley graphs in C 16 × C 16 generated by Marty Malandro. Smith Cayley SRGs 2015 49 / 61

Extending this to larger groups Strongly regular Cayley graphs in C 16 × C 16 generated by Marty Malandro. Smith Cayley SRGs 2015 50 / 61

G = C 32 × C 32 , k = 372 , θ i = 20 , − 12 Smith Cayley SRGs 2015 51 / 61

The rational idempotents graph of C 27 × C 27 Suppose G = C p m × C p m . We define a directed graph on the elements of G/ ∼ by mapping ( x/ ∼ ) �→ ( y/ ∼ ) if x p ∼ y. Equivalence classes of C 27 × C 27 Smith Cayley SRGs 2015 52 / 61

The rational idempotents graph of C 27 × C 27 Suppose G = C p m × C p m . We define a directed graph on the elements of G/ ∼ by mapping ( x/ ∼ ) �→ ( y/ ∼ ) if x p ∼ y. Equivalence classes of C 27 × C 27 Smith Cayley SRGs 2015 52 / 61

Results We have worked through the groups C 2 n × C 2 n up to n = 6 . The size of the automorphism group of the tree grows exponentially, from 3 · 2 22 for C 16 × C 16 to 3 · 2 46 for C 32 × C 32 and then 3 · 2 94 for C 64 × C 64 GAP is quite happy to create those groups, as long as we do not have to look too closely at the elements. But my program hangs up immediately on the second entry of C 32 × C 32 and eventually quits due to memory allocation. Some more theory is needed. If there is some way for me to use the cosets of the automorphism group of the group within the automorphism group of the tree, I can reduce 3 · 2 46 by 3 · 2 13 , so that might search space is 2 33 = 8 billion which is feasible, maybe... Smith Cayley SRGs 2015 53 / 61

Results We have worked through the groups C 2 n × C 2 n up to n = 6 . The size of the automorphism group of the tree grows exponentially, from 3 · 2 22 for C 16 × C 16 to 3 · 2 46 for C 32 × C 32 and then 3 · 2 94 for C 64 × C 64 GAP is quite happy to create those groups, as long as we do not have to look too closely at the elements. But my program hangs up immediately on the second entry of C 32 × C 32 and eventually quits due to memory allocation. Some more theory is needed. If there is some way for me to use the cosets of the automorphism group of the group within the automorphism group of the tree, I can reduce 3 · 2 46 by 3 · 2 13 , so that might search space is 2 33 = 8 billion which is feasible, maybe... Smith Cayley SRGs 2015 53 / 61

Results We have worked through the groups C 2 n × C 2 n up to n = 6 . The size of the automorphism group of the tree grows exponentially, from 3 · 2 22 for C 16 × C 16 to 3 · 2 46 for C 32 × C 32 and then 3 · 2 94 for C 64 × C 64 GAP is quite happy to create those groups, as long as we do not have to look too closely at the elements. But my program hangs up immediately on the second entry of C 32 × C 32 and eventually quits due to memory allocation. Some more theory is needed. If there is some way for me to use the cosets of the automorphism group of the group within the automorphism group of the tree, I can reduce 3 · 2 46 by 3 · 2 13 , so that might search space is 2 33 = 8 billion which is feasible, maybe... Smith Cayley SRGs 2015 53 / 61

Results We have worked through the groups C 2 n × C 2 n up to n = 6 . The size of the automorphism group of the tree grows exponentially, from 3 · 2 22 for C 16 × C 16 to 3 · 2 46 for C 32 × C 32 and then 3 · 2 94 for C 64 × C 64 GAP is quite happy to create those groups, as long as we do not have to look too closely at the elements. But my program hangs up immediately on the second entry of C 32 × C 32 and eventually quits due to memory allocation. Some more theory is needed. If there is some way for me to use the cosets of the automorphism group of the group within the automorphism group of the tree, I can reduce 3 · 2 46 by 3 · 2 13 , so that might search space is 2 33 = 8 billion which is feasible, maybe... Smith Cayley SRGs 2015 53 / 61

Results We have worked through the groups C 2 n × C 2 n up to n = 6 . The size of the automorphism group of the tree grows exponentially, from 3 · 2 22 for C 16 × C 16 to 3 · 2 46 for C 32 × C 32 and then 3 · 2 94 for C 64 × C 64 GAP is quite happy to create those groups, as long as we do not have to look too closely at the elements. But my program hangs up immediately on the second entry of C 32 × C 32 and eventually quits due to memory allocation. Some more theory is needed. If there is some way for me to use the cosets of the automorphism group of the group within the automorphism group of the tree, I can reduce 3 · 2 46 by 3 · 2 13 , so that might search space is 2 33 = 8 billion which is feasible, maybe... Smith Cayley SRGs 2015 53 / 61

Results in C 4 × C 4 In C 4 xC 4 there are, of course, the known PDS: 1 (16 , 6 , 2 , 2) : 2 inequivalent PDS (Rook graph & Shrikhande graphs), each with a different idempotent tree. 2 (16 , 5 , 0 , 2) : 1. (The Clebsch graph is unique.) Smith Cayley SRGs 2015 54 / 61

Results in C 4 × C 4 In C 4 xC 4 there are, of course, the known PDS: 1 (16 , 6 , 2 , 2) : 2 inequivalent PDS (Rook graph & Shrikhande graphs), each with a different idempotent tree. 2 (16 , 5 , 0 , 2) : 1. (The Clebsch graph is unique.) Smith Cayley SRGs 2015 54 / 61

Results in C 4 × C 4 In C 4 xC 4 there are, of course, the known PDS: 1 (16 , 6 , 2 , 2) : 2 inequivalent PDS (Rook graph & Shrikhande graphs), each with a different idempotent tree. 2 (16 , 5 , 0 , 2) : 1. (The Clebsch graph is unique.) Smith Cayley SRGs 2015 54 / 61

Results in C 8 × C 8 There are 11 inequivalent PDS represented by 8 different idempotent trees. 1 (64 , 14 , 6 , 2) : 1 2 (64 , 18 , 2 , 6) : 1 3 (64 , 21 , 8 , 6) : 3 inequivalent PDS in 2 idempotent trees, 4 Hadamard: (64 , 27 , 10 , 12) : 3 inequivalent PDS in 2 idempotent trees, 5 Hadamard: (64 , 28 , 12 , 12) : 3 inequivalent PDS in 2 idempotent trees. Smith Cayley SRGs 2015 55 / 61