Equiangular lines in Euclidean spaces

Size: px

Start display at page:

Download "Equiangular lines in Euclidean spaces"

Wesley Ward
6 years ago
Views:

1 Equiangular lines in Euclidean spaces Gary Greaves 東北大学 Tohoku University 14th August 215 joint work with J. Koolen, A. Munemasa, and F. Szöllősi. Gary Greaves Equiangular lines in Euclidean spaces 1/23

2 Plan From lines to matrices; Upper bounds for N(d); Lower bounds for N(d); Seidel matrices with 3 eigenvalues; A strengthening of the relative bound; New upper bounds. Gary Greaves Equiangular lines in Euclidean spaces 2/23

3 Equiangular line systems Let L be a system of n lines spanned by v 1,..., v n R d. L is equiangular if v i, v i = 1 and v i, v j = α (α is called the common angle). Problem: given d, what is the largest possible number N(d) of equiangular lines in R d? Questions Martin (215): Find, as many as you can, equiangular lines in R d. Yu (215): Determine N(14) and N(16). Gary Greaves Equiangular lines in Euclidean spaces 3/23

4 Seidel matrices Let L be an equiangular line system of n lines in R d with common angle α. Let M be the Gram matrix for the line system L. Then M is positive semidefinite with nullity n d. Assume α > and set S = (M I)/α. S is a {, ±1}-matrix with smallest eigenvalue 1/α with multiplicity n d. S = S(L) is called a Seidel matrix. Relation to graphs: S = J I 2A. Gary Greaves Equiangular lines in Euclidean spaces 4/23

5 Seidel matrices Let L be an equiangular line system of n lines in R d with common angle α. Let M be the Gram matrix for the line system L. Then M is positive semidefinite with nullity n d. Assume α > and set S = (M I)/α. S is a {, ±1}-matrix with smallest eigenvalue 1/α with multiplicity n d. Smallest eigenvalue λ = 1/α. S = S(L) is called a Seidel matrix. Relation to graphs: S = J I 2A. Gary Greaves Equiangular lines in Euclidean spaces 4/23

6 Basic properties of Seidel matrices Let S be an n n Seidel matrix. tr S =, tr S 2 = n(n 1); GG, AM, JK, FS (215+): det S 4 ( 1) n 1 (n 1); Neumann (1973): If S has an even integer eigenvalue λ then λ has multiplicity at most 1. 2A = J I S. Gary Greaves Equiangular lines in Euclidean spaces 5/23

7 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

8 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

9 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

10 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

11 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

12 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

13 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

14 Icosahedron o o1 i o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves Equiangular lines in Euclidean spaces 6/23

15 Icosahedron S = ; Spectrum: {[ 5] 3, [ 5] 3 }; n = 6, d = 3, and α = 1/ 5. Question: for d = 3, can we do better than n = 6? Gary Greaves Equiangular lines in Euclidean spaces 6/23

16 Upper bounds for N(d) Let L be an equiangular line system of n lines in R d whose Seidel matrix has smallest eigenvalue λ. Gerzon (1973): Absolute bound: n d(d + 1). 2 Van Lint and Seidel (1966): for λ 2 d + 2 Relative bound: n d(λ2 1) λ 2 d. Neumann (1973): If n > 2d then λ is an odd integer. Gary Greaves Equiangular lines in Euclidean spaces 7/23

17 Upper bounds for N(d) Let L be an equiangular line system of n lines in R d whose Seidel matrix has smallest eigenvalue λ. Van Lint and Seidel (1966): for λ 2 d + 2 Relative bound: n d(λ2 1) λ 2 d. Neumann (1973): If n > 2d then λ is an odd integer. Barg and Yu (213): SDP upper bounds for d 136. Okuda and Yu (215+): New relative bound. Gary Greaves Equiangular lines in Euclidean spaces 7/23

18 Lower Bounds Lemmens and Seidel (1973): N(d) d d. de Caen (2): N(d) m 2 /2 for d = 3m/2 1 (m = 4 t ). Szöllősi (211): = N(d) (d + 2) 2 /72. Gary Greaves Equiangular lines in Euclidean spaces 8/23

19 Lower Bounds Lemmens and Seidel (1973): N(d) d d. de Caen (2): N(d) m 2 /2 for d = 3m/2 1 (m = 4 t ). Szöllősi (211): = N(d) (d + 2) 2 /72. GG, AM, JK, FS (215+): N(d) 32d2 +328d Gary Greaves Equiangular lines in Euclidean spaces 8/23

20 Lower Bounds de Caen (2): N(d) m 2 /2 for d = 3m/2 1 (m = 4 t ). Szöllősi (211): = N(d) (d + 2) 2 /72. GG, AM, JK, FS (215+): N(d) 32d2 +328d Proposition (GG, AM, JK, FS 215+) For each t 1 and m = 4 t there exists an equiangular set of (a) m(m/2 + 1) lines in dimension R 3m/2+1 ; and (b) m(m/2 + 1) 1 lines in dimension R 3m/2 ; and (c) mj lines in dimension R m+j 1 for every 1 j m/2. Gary Greaves Equiangular lines in Euclidean spaces 8/23

21 Illustration of the proof Let m = 4. There is a set of m/2 + 1 MUBs of R , , Gary Greaves Equiangular lines in Euclidean spaces 9/23

22 Illustration of the proof Let m = 4. There is a set of m/2 + 1 MUBs of R , , Gary Greaves Equiangular lines in Euclidean spaces 9/23

23 Illustration of the proof Let m = 4. There is a set of m/2 + 1 MUBs of R , , Gary Greaves Equiangular lines in Euclidean spaces 9/23

24 Illustration of the proof We have 12 lines in R 7 ; Gary Greaves Equiangular lines in Euclidean spaces 9/23

25 Illustration of the proof We have 12 lines in R 7 ; We have 11 lines in R 7 ; Gary Greaves Equiangular lines in Euclidean spaces 9/23

26 Illustration of the proof We have 12 lines in R 7 ; We have 11 lines in R 6 ; Left nullspace: Gary Greaves Equiangular lines in Euclidean spaces 9/23

27 Illustration of the proof We have 12 lines in R 7 ; We have 11 lines in R 6 ; We have 8 lines in R 6 ; Left nullspace: Gary Greaves Equiangular lines in Euclidean spaces 9/23

28 Illustration of the proof We have 12 lines in R 7 ; We have 11 lines in R 6 ; We have 8 lines in R 5 ; Left nullspace: 2, Gary Greaves Equiangular lines in Euclidean spaces 9/23

29 Illustration of the proof We have 12 lines in R 7 ; We have 11 lines in R 6 ; We have 8 lines in R 5 ; Left nullspace: < Same as de Caen 2, Gary Greaves Equiangular lines in Euclidean spaces 9/23

30 Seidel matrices with few eigenvalues Gary Greaves Equiangular lines in Euclidean spaces 1/23

31 Seidel matrices with two distinct eigenvalues Let L be an equiangular line system of n lines in R d with common angle α. If n meets either the absolute bound or the relative bound then S(L) has precisely two distinct eigenvalues. Related to strongly regular graphs (regular graphs with 3 eigenvalues). Gary Greaves Equiangular lines in Euclidean spaces 11/23

32 Seidel matrices with two distinct eigenvalues Let L be an equiangular line system of n lines in R d with common angle α. If n meets either the absolute bound or the relative bound then S(L) has precisely two distinct eigenvalues. Related to strongly regular graphs (regular graphs with 3 eigenvalues). Seidel (1995): d n Gary Greaves Equiangular lines in Euclidean spaces 11/23

33 Seidel matrices with two distinct eigenvalues Let L be an equiangular line system of n lines in R d with common angle α. If n meets either the absolute bound or the relative bound then S(L) has precisely two distinct eigenvalues. Related to strongly regular graphs (regular graphs with 3 eigenvalues). d n Gary Greaves Equiangular lines in Euclidean spaces 11/23

34 Seidel matrices with two distinct eigenvalues Let L be an equiangular line system of n lines in R d with common angle α. If n meets either the absolute bound or the relative bound then S(L) has precisely two distinct eigenvalues. Related to strongly regular graphs (regular graphs with 3 eigenvalues). d n Gary Greaves Equiangular lines in Euclidean spaces 11/23

35 Seidel matrices with two distinct eigenvalues Let L be an equiangular line system of n lines in R d with common angle α. If n meets either the absolute bound or the relative bound then S(L) has precisely two distinct eigenvalues. Related to strongly regular graphs (regular graphs with 3 eigenvalues). d n Gary Greaves Equiangular lines in Euclidean spaces 11/23

36 Seidel matrices with two distinct eigenvalues Let L be an equiangular line system of n lines in R d with common angle α. If n meets either the absolute bound or the relative bound then S(L) has precisely two distinct eigenvalues. Related to strongly regular graphs (regular graphs with 3 eigenvalues). d n Gary Greaves Equiangular lines in Euclidean spaces 11/23

37 Seidel matrices with two distinct eigenvalues Let L be an equiangular line system of n lines in R d with common angle α. If n meets either the absolute bound or the relative bound then S(L) has precisely two distinct eigenvalues. Related to strongly regular graphs (regular graphs with 3 eigenvalues). d n Gary Greaves Equiangular lines in Euclidean spaces 11/23

38 Seidel matrices with 3 eigenvalues: constructions SRG construction Let A be the adjacency matrix of an n-vertex strongly regular graph with spectrum {[k] 1, [θ] a, [τ] b }. S = J I 2A has at most 3 distinct eigenvalues: {[n 1 2k] 1, [ 1 2θ] a, [ 1 2τ] b }; Gary Greaves Equiangular lines in Euclidean spaces 12/23

39 Seidel matrices with 3 eigenvalues: constructions SRG construction Let A be the adjacency matrix of an n-vertex strongly regular graph with spectrum {[k] 1, [θ] a, [τ] b }. S = J I 2A has at most 3 distinct eigenvalues: {[n 1 2k] 1, [ 1 2θ] a, [ 1 2τ] b }; Tensor construction Let S be an n n Seidel matrix with spectrum {[λ ] n d, [λ 1 ] d }. J m (S I n ) + I nm has at most 3 distinct eigenvalues: {[m(λ 1) + 1] n d, [1] n(m 1), [m(λ 1 1) + 1] d }. Gary Greaves Equiangular lines in Euclidean spaces 12/23

40 From two eigenvalues to three: coclique removal Let S be an n n Seidel matrix with spectrum {[λ ] n d, [λ 1 ] d }. Take a graph G in the switching class of S. For any vertex v we have the following lemma. Lemma The Seidel matrix corresponding to G\{v} has spectrum {[λ ] n d 1, [λ 1 ] d 1, [λ + λ 1 ] 1 }. Gary Greaves Equiangular lines in Euclidean spaces 13/23

41 From two eigenvalues to three: coclique removal Let S be an n n Seidel matrix with spectrum {[λ ] n d, [λ 1 ] d }. Take a graph G in the switching class of S. For any vertex v we have the following lemma. Lemma The Seidel matrix corresponding to G\{v} has spectrum {[λ ] n d 1, [λ 1 ] d 1, [λ + λ 1 ] 1 }. Gary Greaves Equiangular lines in Euclidean spaces 13/23

42 From two eigenvalues to three: coclique removal Let S be an n n Seidel matrix with spectrum {[λ ] n d, [λ 1 ] d }. Suppose that a graph G in the switching class of S that contains a coclique C on c min(n d, d) vertices. Theorem (GG, AM, JK, FS 215+) The Seidel matrix corresponding to G\C has spectrum {[λ ] n d c, [λ 1 ] d c, [λ + λ c] 1, [λ + λ 1 + 1] c 1 }. Gary Greaves Equiangular lines in Euclidean spaces 13/23

43 From two eigenvalues to three: coclique removal Theorem (GG, AM, JK, FS 215+) The Seidel matrix corresponding to G\C has spectrum {[λ ] n d c, [λ 1 ] d c, [λ + λ c] 1, [λ + λ 1 + 1] c 1 }. Example (cf. Tremain 28) Start with Seidel matrix S with spectrum {[ 5] 21, [7] 15 }. spec(s\k 1 ) = {[ 5] 2, [2] 1, [7] 14 } spec(s\k 2 ) = {[ 5] 19, [1] 1, [3] 1, [7] 13 } spec(s\k 3 ) = {[ 5] 18, [] 1, [3] 2, [7] 12 } spec(s\k 4 ) = {[ 5] 17, [ 1] 1, [3] 3, [7] 11 } spec(s\k 5 ) = {[ 5] 16, [ 2] 1, [3] 4, [7] 1 } spec(s\k 6 ) = {[ 5] 15, [ 3] 1, [3] 5, [7] 9 } spec(s\k 7 ) = {[ 5] 14, [ 4] 1, [3] 6, [7] 8 } spec(s\k 8 ) = {[ 5] 13, [ 5] 1, [3] 7, [7] 7 } Gary Greaves Equiangular lines in Euclidean spaces 13/23

44 From two eigenvalues to three: coclique removal Theorem (GG, AM, JK, FS 215+) The Seidel matrix corresponding to G\C has spectrum {[λ ] n d c, [λ 1 ] d c, [λ + λ c] 1, [λ + λ 1 + 1] c 1 }. Example (cf. Tremain 28) Start with Seidel matrix S with spectrum {[ 5] 21, [7] 15 }. spec(s\k 1 ) = {[ 5] 2, [2] 1, [7] 14 } spec(s\k 2 ) = {[ 5] 19, [1] 1, [3] 1, [7] 13 } spec(s\k 3 ) = {[ 5] 18, [] 1, [3] 2, [7] 12 } spec(s\k 4 ) = {[ 5] 17, [ 1] 1, [3] 3, [7] 11 } spec(s\k 5 ) = {[ 5] 16, [ 2] 1, [3] 4, [7] 1 } spec(s\k 6 ) = {[ 5] 15, [ 3] 1, [3] 5, [7] 9 } spec(s\k 7 ) = {[ 5] 14, [ 4] 1, [3] 6, [7] 8 } spec(s\k 8 ) = {[ 5] 13, [ 5] 1, [3] 7, [7] 7 } = {[ 5] 14, [3] 7, [7] 7 } Gary Greaves Equiangular lines in Euclidean spaces 13/23

45 From two eigenvalues to three: coclique removal Theorem (GG, AM, JK, FS 215+) The Seidel matrix corresponding to G\C has spectrum {[λ ] n d c, [λ 1 ] d c, [λ + λ c] 1, [λ + λ 1 + 1] c 1 }. Resulting Seidel matrix has at most 3 eigenvalues when c = λ or c = d. Only two eigenvalues when c = d = λ This technique is useful if we know something about the size of a coclique in G. Gary Greaves Equiangular lines in Euclidean spaces 13/23

46 Properties of Seidel matrices with 3 eigenvalues Theorem (GG, AM, JK, FS 215+) Let S be an n n Seidel matrix with precisely 3 distinct eigenvalues. Then S has a rational eigenvalue. Furthermore, if n 3 mod 4 then every eigenvalue of S is rational. Gary Greaves Equiangular lines in Euclidean spaces 14/23

47 Properties of Seidel matrices with 3 eigenvalues Theorem (GG, AM, JK, FS 215+) Let S be an n n Seidel matrix with precisely 3 distinct eigenvalues. Then S has a rational eigenvalue. Furthermore, if n 3 mod 4 then every eigenvalue of S is rational. Theorem (GG, AM, JK, FS 215+) For primes p 3 mod 4, there do not exist any p p Seidel matrices having precisely 3 distinct eigenvalues. Except for n = 4, they exist for all other n. n # Gary Greaves Equiangular lines in Euclidean spaces 14/23

48 Strengthening the relative bound Gary Greaves Equiangular lines in Euclidean spaces 15/23

49 Strengthening the relative bound Theorem (Relative bound) Let L be an equiangular line system of n lines in R d whose Seidel matrix has smallest eigenvalue λ and suppose λ 2 d + 2. n d(λ2 1) λ 2 d. Equality implies that S has 2 distinct eigenvalues. Gary Greaves Equiangular lines in Euclidean spaces 16/23

50 Strengthening the relative bound Theorem (Relative bound) Let L be an equiangular line system of n lines in R d whose Seidel matrix has smallest eigenvalue λ and suppose λ 2 d + 2. n d(λ2 1) λ 2 d. Equality implies that S has 2 distinct eigenvalues. GG, AM, JK, FS (215+): Suppose n = and there exists µ Z satisfying d(λ 2 1) λ 2 d certain conditions. Then S has at most 4 distinct eigenvalues. {[λ ] n d, [µ 1] a, [µ] b, [µ + 1] d a b }. Gary Greaves Equiangular lines in Euclidean spaces 16/23

51 Case study: equiangular lines in R 14 Suppose there is n > 2 14 equiangular lines in R 14. Neumann (1973): λ is an odd integer. Lemmens and Seidel (1973): N 3 (14) = 28. Relative bound = λ = 5, N 5 (14) Suppose we have n = 3 (d = 14), with corresponding Seidel matrix S having eigenvalues λ (n d) < λ 1 λ 2 λ d. Gary Greaves Equiangular lines in Euclidean spaces 17/23

52 Case study: equiangular lines in R 14 Lemmens and Seidel (1973): N 3 (14) = 28. Relative bound = λ = 5, N 5 (14) Suppose we have n = 3 (d = 14), with corresponding Seidel matrix S having eigenvalues λ (n d) < λ 1 λ 2 λ d. Using the trace formulae, we have d λ i = (n d)λ = 8; i=1 d λ 2 i = n(n 1) (n d)λ 2 = 47. i=1 Gary Greaves Equiangular lines in Euclidean spaces 17/23

53 Case study: equiangular lines in R 14 Suppose we have n = 3 (d = 14), with corresponding Seidel matrix S having eigenvalues λ (n d) < λ 1 λ 2 λ d. Using the trace formulae, we have Then d λ i = (n d)λ = 8; i=1 d λ 2 i = n(n 1) (n d)λ 2 = 47. i=1 1 = d i=1 (λ i 6) 2 /d Gary Greaves Equiangular lines in Euclidean spaces 17/23

54 Case study: equiangular lines in R 14 Suppose we have n = 3 (d = 14), with corresponding Seidel matrix S having eigenvalues λ (n d) < λ 1 λ 2 λ d. Using the trace formulae, we have Then d λ i = (n d)λ = 8; i=1 d λ 2 i = n(n 1) (n d)λ 2 = 47. i=1 1 = d i=1 (λ i 6) 2 /d d (λ i 6) 2 1. Gary Greaves Equiangular lines in Euclidean spaces 17/23

55 Case study: equiangular lines in R 14 Suppose we have n = 3 (d = 14), with corresponding Seidel matrix S having eigenvalues λ (n d) < λ 1 λ 2 λ d. Using the trace formulae, we have Then d λ i = (n d)λ = 8; i=1 d λ 2 i = n(n 1) (n d)λ 2 = 47. i=1 1 = d i=1 Hence (λ i 6) {±1}. (λ i 6) 2 /d d (λ i 6) 2 1. Gary Greaves Equiangular lines in Euclidean spaces 17/23

56 Relative bound in low dimensions d λ d(λ 2 1) λ 2 d Spectrum {[ 5] 16, [5] 9, [7] 5 } {[ 5] 21, [7] 15 } {[ 5] 26, [7] 7, [9] 9 } {[ 5] 34, [1] 17 } {[ 5] 43, [11] 9, [12] 1, [13] 8 } {[ 5] 57, [15] 19 } {[ 5] 76, [19] 2 } Gary Greaves Equiangular lines in Euclidean spaces 18/23

57 Relative bound in low dimensions d λ d(λ 2 1) λ 2 d Spectrum {[ 5] 16, [5] 9, [7] 5 } {[ 5] 21, [7] 15 } {[ 5] 26, [7] 7, [9] 9 } {[ 5] 34, [1] 17 } {[ 5] 43, [11] 9, [12] 1, [13] 8 } {[ 5] 57, [15] 19 } {[ 5] 76, [19] 2 } Gary Greaves Equiangular lines in Euclidean spaces 18/23

58 Relative bound in low dimensions d λ d(λ 2 1) λ 2 d Spectrum {[ 5] 16, [5] 9, [7] 5 } {[ 5] 21, [7] 15 } {[ 5] 26, [7] 7, [9] 9 } {[ 5] 34, [1] 17 } {[ 5] 43, [11] 9, [12] 1, [13] 8 } {[ 5] 57, [15] 19 } {[ 5] 76, [19] 2 } Seidel matrices cannot have even eigenvalues with multiplicity greater than 1. 2A = J I S. Gary Greaves Equiangular lines in Euclidean spaces 18/23

59 Relative bound in low dimensions d λ d(λ 2 1) λ 2 d Spectrum {[ 5] 16, [5] 9, [7] 5 } {[ 5] 21, [7] 15 } {[ 5] 26, [7] 7, [9] 9 } {[ 5] 34, [1] 17 } {[ 5] 43, [11] 9, [12] 1, [13] 8 } {[ 5] 57, [15] 19 } {[ 5] 76, [19] 2 } Seidel matrices cannot have even eigenvalues with multiplicity greater than 1. 2A = J I S. Gary Greaves Equiangular lines in Euclidean spaces 18/23

60 Euler graphs An Euler graph is a graph each of whose vertices have even valency. Theorem (Mallows-Sloane 1975) The number of switching classes of n n Seidel matrices equals the number of Euler graphs on n vertices. Theorem (GG, AM, JK, FS 215+) Let S be a Seidel matrix with precisely 3 distinct eigenvalues. Then S is switching equivalent to a Seidel matrix S = J I 2A where A is the adjacency matrix of an Euler graph. Gary Greaves Equiangular lines in Euclidean spaces 19/23

61 Killing Seidel matrices with three eigenvalues Theorem (GG, AM, JK, FS 215+) Let S be an n n Seidel matrix with spec. {[λ] a, [µ] b, [ν] c }. Suppose n 2 mod 4, λ + µ mod 4, and n 1 + λµ = 4. Then ν 2 (λ + µ)ν + λµ /4 = n/c Z and ν n/c 1. Gary Greaves Equiangular lines in Euclidean spaces 2/23

62 Killing Seidel matrices with three eigenvalues Theorem (GG, AM, JK, FS 215+) Let S be an n n Seidel matrix with spec. {[λ] a, [µ] b, [ν] c }. Suppose n 2 mod 4, λ + µ mod 4, and n 1 + λµ = 4. Then ν 2 (λ + µ)ν + λµ /4 = n/c Z and ν n/c 1. Corollary The candidate Seidel matrices with spectra {[ 5] 16, [5] 9, [7] 5 } and {[ 5] 26, [7] 7, [9] 9 } do not exist. Corollary Regular graphs with spectra {[11] 1, [2] 16, [ 3] 9, [ 4] 4 } and {[12] 1, [2] 16, [ 3] 8, [ 4] 5 } do not exist. Gary Greaves Equiangular lines in Euclidean spaces 2/23

63 Case study: equiangular lines in R 14 Let S be the Seidel matrix with spectrum {[ 5] 16, [5] 9, [7] 5 }. The matrix M := S 2 25I has spectrum {[] 25, [24] 5 }. M = 28J 24I 2(AJ + JA) + 4(A 2 + A) mod 4. M is positive semidefinite and M ii = 4 = M ij {, ±4}. Deduce M is switching equivalent to 4J 6 I 5. Evaluating [S 3 ] 11 in two different ways gives the bound. Gary Greaves Equiangular lines in Euclidean spaces 21/23

64 Conclusion N(14) {28, 29} and N(16) {4, 41}. Show that 3 (42) equiangular lines in R 14 (R 16 ) implies that the associated Seidel matrix has spectrum {[ 5] 16, [5] 9, [7] 5 } ({[ 5] 26, [7] 7, [9] 9 }). Kill the corresponding Seidel matrices with three eigenvalues. Can we construct or kill other Seidel matrices with three eigenvalues? What about {[ 5] 43, [11] 9, [12] 1, [13] 8 }? (N(18) = 61?) Gary Greaves Equiangular lines in Euclidean spaces 22/23

65 Feasible Seidel matrices with 3 eigenvalues n d λ µ ν Existence Remark [ 5] 14 [3] 7 [7] 7 Y 3 14 [ 5] 16 [5] 9 [7] 5 N 4 16 [ 5] 24 [5] 6 [9] 1? 4 16 [ 5] 24 [7] 15 [15] 1 Y [ 5] 26 [7] 7 [9] 9 N [ 5] 31 [7] 8 [11] 9 Y [ 5] 32 [9] 16 [16] 1? [ 5] 3 [3] 6 [11] 12? [ 5] 3 [7] 16 [19] 2? [ 5] 36 [7] 9 [13] 9? 6 18 [ 5] 42 [11] 15 [15] 3? coclique removal? [ 5] 53 [13] 16 [19] 3 Y [ 5] 56 [1] 1 [15] 18? from {[ 5] 57, [15] 19 } 9 2 [ 5] 7 [13] 5 [19] 15? 95 2 [ 5] 75 [14] 1 [19] 19? from {[ 5] 76, [19] 2 } Gary Greaves Equiangular lines in Euclidean spaces 23/23

Graphs With Many Valencies and Few Eigenvalues

Graphs With Many Valencies and Few Eigenvalues By Edwin van Dam, Jack H. Koolen, Zheng-Jiang Xia ELA Vol 28. 2015 Vicente Valle Martinez Math 595 Spectral Graph Theory Apr 14, 2017 V. Valle Martinez (Math