601 research outputs found
Largest regular multigraphs with three distinct eigenvalues
We deal with connected -regular multigraphs of order that has only
three distinct eigenvalues. In this paper, we study the largest possible number
of vertices of such a graph for given . For , the Moore graphs are
largest. For , we show an upper bound , with
equality if and only if there exists a finite projective plane of order
that admits a polarity.Comment: 9 pages, no figur
A generalization of Larman-Rogers-Seidel's theorem
A finite set X in the d-dimensional Euclidean space is called an s-distance
set if the set of Euclidean distances between any two distinct points of X has
size s. Larman--Rogers--Seidel proved that if the cardinality of a two-distance
set is greater than 2d+3, then there exists an integer k such that
a^2/b^2=(k-1)/k, where a and b are the distances. In this paper, we give an
extension of this theorem for any s. Namely, if the size of an s-distance set
is greater than some value depending on d and s, then certain functions of s
distances become integers. Moreover, we prove that if the size of X is greater
than the value, then the number of s-distance sets is finite.Comment: 12 pages, no figur
Complex spherical codes with two inner products
A finite set in a complex sphere is called a complex spherical -code
if the number of inner products between two distinct vectors in is equal to
. In this paper, we characterize the tight complex spherical -codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric -optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.Comment: 10 pages, to appear in European Journal of Combinatoric
A characterization of skew Hadamard matrices and doubly regular tournaments
We give a new characterization of skew Hadamard matrices of size in terms
of the data of the spectra of tournaments of size .Comment: 9 page
Complex spherical codes with three inner products
Let be a finite set in a complex sphere of dimension. Let be
the set of usual inner products of two distinct vectors in . A set is
called a complex spherical -code if the cardinality of is and
contains an imaginary number. We would like to classify the largest
possible -codes for given dimension . In this paper, we consider the
problem for the case . Roy and Suda (2014) gave a certain upper bound for
the cardinalities of -codes. A -code is said to be tight if
attains the bound. We show that there exists no tight -code except for
dimensions , . Moreover we make an algorithm to classify the largest
-codes by considering representations of oriented graphs. By this algorithm,
the largest -codes are classified for dimensions , , with a
current computer.Comment: 26 pages, no figur
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