Equiangular lines in Euclidean spaces

A class of non-oriented simple graphs is called Seidel switching self-complementary if the complement of any representing graph is in the same equivalence class. This note introduces the 3-signature (s, t) of a switching class of n-vertex graphs. The numbers s and t are the numbers of positive and negative triples within any representing graph of the class. It appears that, for any switching self-complementary class of n-vertex graphs, these numbers are equal, yielding n 3 even. Consequently if n ≡ 3 (mod 4) then there is no switching self-complementary class of n-vertex graphs. We then consider the switching classes of Paley conference graphs with 4k + 2 vertices, 4k + 1 = p α , p an odd prime and α a positive integer. We reprove that these classes are switching selfcomplementary. Moreover, it is proven that all 4k-vertex graphs contained in a (4k + 2)-vertex Paley conference graph are switching equivalent and their class is still a switching self-complementary class. In addition, the 3-signature is generalized in view of obtaining a complete invariant of switching classes up to order 8.

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“…Greaves, Koolen, Munemasa and Szöllösi [7] obtained the same result (Proposition 3.7) using different methods.…”

Section: Theorem 1 If a Switching Class Of N-vertex Graphs Is A Sssupporting

confidence: 61%

On switching classes of graphs

Et-Taoui

Fruchard

2018

Linear Algebra and its Applications

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“…The above discussion proves the following theorem: For n = 22 and each of the intersection patterns z z z ( , , ) = (12, 12, 0), 3 4 5 (13,8,3), (14,4,6), the system of Equations (20) and (21) is the same. This system has the solutions x x x ( , , ) = (1, 3, 6) 11 12 22 and x x x ( , , ) = (2, 6, 2) 11 12 22 .…”

Section: By Solving This System Of Equations We Obtainmentioning

confidence: 68%

“…A computer search showed that there exists no Seidel matrix of order

13

with smallest eigenvalue equal to

goodbreakinfix- 5

with multiplicity

4

. This proved that

B^{'} (10, 2) goodbreakinfix= 12

.…”

Section: For Ngoodbreakinfix>10 Ngoodbreakinfix+1goodbreakinfix=boldmentioning

confidence: 97%

The maximum number of columns in supersaturated designs with

Morales

Bulutoglu

Arasu

2019

J of Combinatorial Designs

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Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns B ( n , t ) that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows ( n) and a maximum in absolute value correlation between any two columns ( t ∕ n). In particular, they proved that B ( n , 2 ) goodbreakinfix≤ n goodbreakinfix+ 2 for n goodbreakinfix≡ 2 (mod 4) and n goodbreakinfix> 6. However, the only known SSD satisfying this upper bound is when n goodbreakinfix= 10. By utilizing a computer search, we prove that B ( n , 2 ) goodbreakinfix≤ n goodbreakinfix+ 1 for n goodbreakinfix= 18 goodbreakinfix, 22 goodbreakinfix, 30, and B ( 14 , 2 ) goodbreakinfix= 15. These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the E ( s 2 ) lower bound for SSDs with n rows and n goodbreakinfix+ 2 columns, for n goodbreakinfix= 14 goodbreakinfix, 18 goodbreakinfix, 22, and 30. Finally, we show that a skew‐type Hadamard matrix of order n can be used to construct an SSD with n goodbreakinfix− 2 rows and n goodbreakinfix− 1 columns that proves B ( n − 2 , 2 ) goodbreakinfix≥ n goodbreakinfix− 1. Hence, we establish B ( n , 2 ) goodbreakinfix= n goodbreakinfix+ 1 for n goodbreakinfix= 14 goodbreakinfix, 18 goodbreakinfix, 22 goodbreakinfix, 30 and B ( n , 2 ) goodbreakinfix≥ n goodbreakinfix+ 1 for all n goodbreakinfix≡ 2 (mod 4) such that n goodbreakinfix≤ 270. Our result also implies that B ( n , 2 ) goodbreakinfix≥ n goodbreakinfix+ 1 when n goodbreakinfix+ 1 is a prime power and n goodbreakinfix+ 1 goodbreakinfix≡ 3 (mod 4). We conjecture that n goodbreakinfix+ 1 goodbreakinfix= B ( n , 2 ) goodbreakinfix< B ′ ( n , 2 ) goodbreakinfix= n goodbreakinfix+ 2 for all n goodbreakinfix> 10 and n goodbreakinfix≡ 2 (mod 4), where B ′ ( n , 2 ) is the maximum number of equiangular lines in R n − 1 with pairwise angle arccos ( 2 ∕ n ).

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“…Proof. Let M = (4t + 3)I + S 2 and let P ∈ O 4t+1 (Z) satisfy (9). We write r i for the ith distinct row of P MP .…”

Section: Proof For the Skew-symmetric Casementioning

confidence: 99%

“…Lemma 5.24 of[9]). Let M be a positive semidefinite {0, ±1}-matrix with constant diagonal entries 1.…”

mentioning

confidence: 99%