Equiangular lines in Euclidean spaces

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

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

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

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

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

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

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

Icosahedron 1 1 1 1 1 1 1 1 1 1 S = 1 1 1 1 1 1 1 1 1 1 ; 1 1 1 1 1 1 1 1 1 1 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

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

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

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

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+296 189. Gary Greaves Equiangular lines in Euclidean spaces 8/23

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+296 189. 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

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

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

Illustration of the proof Let m = 4. There is a set of m/2 + 1 MUBs of R 4. 1 1 1 1 1 1 1 1 1 1 1, 1 1 1 1 1 2 1 1 1 1, 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 9/23

Illustration of the proof We have 12 lines in R 7 ; 1 6 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 9/23

Illustration of the proof We have 12 lines in R 7 ; We have 11 lines in R 7 ; 1 6 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 9/23

Illustration of the proof We have 12 lines in R 7 ; We have 11 lines in R 6 ; Left nullspace: 2 1 1 1 6 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 9/23

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: 2 1 1 1 6 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 9/23

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, 1 1 1 1 6 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 9/23

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, 1 1 1 1 6 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 9/23

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

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

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 2 3 4 5 6 7 13 14 15 16 17 18 19 2 n 3 6 6 1 16 28 28 36 4 48 48 72 9 76 96 Gary Greaves Equiangular lines in Euclidean spaces 11/23

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 2 3 4 5 6 7 13 14 15 16 17 18 19 2 n 3 6 6 1 16 28 28 36 4 48 48 72 9 3 42 5 61 76 96 Gary Greaves Equiangular lines in Euclidean spaces 11/23

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 2 3 4 5 6 7 13 14 15 16 17 18 19 2 n 3 6 6 1 16 28 28 36 4 48 48 72 9 3 42 5 61 76 96 2 2 2 2 2 2 2 2 Gary Greaves Equiangular lines in Euclidean spaces 11/23

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 2 3 4 5 6 7 13 14 15 16 17 18 19 2 n 3 6 6 1 16 28 28 36 4 48 48 72 9 3 42 5 61 76 96 2 2 2 2 2 2 2 2 3 3 3 3 3 3 Gary Greaves Equiangular lines in Euclidean spaces 11/23

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 2 3 4 5 6 7 13 14 15 16 17 18 19 2 n 3 6 6 1 16 28 28 36 4 48 48 72 9 3 42 5 61 76 96 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 Gary Greaves Equiangular lines in Euclidean spaces 11/23

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 2 3 4 5 6 7 13 14 15 16 17 18 19 2 n 3 6 6 1 16 28 28 36 4 48 48 72 9 29 41 5 61 76 96 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 Gary Greaves Equiangular lines in Euclidean spaces 11/23

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

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

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 1 + 1 c] 1, [λ + λ 1 + 1] c 1 }. Gary Greaves Equiangular lines in Euclidean spaces 13/23

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 1 + 1 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

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 1 + 1 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

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 1 + 1 c] 1, [λ + λ 1 + 1] c 1 }. Resulting Seidel matrix has at most 3 eigenvalues when c = λ 1 + 1 or c = d. Only two eigenvalues when c = d = λ 1 + 1. 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

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 3 4 5 6 7 8 9 1 11 12 # 1 2 2 3 4 1 Gary Greaves Equiangular lines in Euclidean spaces 14/23

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

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

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) 3.54.... 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

Case study: equiangular lines in R 14 Lemmens and Seidel (1973): N 3 (14) = 28. Relative bound = λ = 5, N 5 (14) 3.54.... 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

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

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

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

Relative bound in low dimensions d λ d(λ 2 1) λ 2 d Spectrum 14 5 3 {[ 5] 16, [5] 9, [7] 5 } 15 5 36 {[ 5] 21, [7] 15 } 16 5 42 {[ 5] 26, [7] 7, [9] 9 } 17 5 51 {[ 5] 34, [1] 17 } 18 5 61 {[ 5] 43, [11] 9, [12] 1, [13] 8 } 19 5 76 {[ 5] 57, [15] 19 } 2 5 96 {[ 5] 76, [19] 2 } Gary Greaves Equiangular lines in Euclidean spaces 18/23

Relative bound in low dimensions d λ d(λ 2 1) λ 2 d Spectrum 14 5 3 {[ 5] 16, [5] 9, [7] 5 } 15 5 36 {[ 5] 21, [7] 15 } 16 5 42 {[ 5] 26, [7] 7, [9] 9 } 17 5 51 {[ 5] 34, [1] 17 } 18 5 61 {[ 5] 43, [11] 9, [12] 1, [13] 8 } 19 5 76 {[ 5] 57, [15] 19 } 2 5 96 {[ 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

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

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

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

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

Feasible Seidel matrices with 3 eigenvalues n d λ µ ν Existence Remark 28 14 [ 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 42 16 [ 5] 26 [7] 7 [9] 9 N 48 17 [ 5] 31 [7] 8 [11] 9 Y 49 17 [ 5] 32 [9] 16 [16] 1? 48 18 [ 5] 3 [3] 6 [11] 12? 48 18 [ 5] 3 [7] 16 [19] 2? 54 18 [ 5] 36 [7] 9 [13] 9? 6 18 [ 5] 42 [11] 15 [15] 3? coclique removal? 72 19 [ 5] 53 [13] 16 [19] 3 Y 75 19 [ 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