Largest regular multigraphs with three distinct eigenvalues

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k\ne 2,3,7,57$, we show an upper bound $n\leq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ 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 $X$ in a complex sphere is called a complex spherical $2$-code if the number of inner products between two distinct vectors in $X$ is equal to $2$. In this paper, we characterize the tight complex spherical $2$-codes by doubly regular tournaments, or skew Hadamard matrices. We also give certain maximal 2-codes relating to skew-symmetric $D$-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 $n$ in terms of the data of the spectra of tournaments of size $n-2$.Comment: 9 page

On a generalization of distance sets

A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distinct distances between two distinct points in $X$ and a subset $X$ is called a locally $k$-distance set if for any point $x$ in $X$, there are at most $k$ distinct distances between $x$ and other points in $X$. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of $k$-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally $k$-distance sets on a sphere. In the first part of this paper, we prove that if $X$ is a locally $k$-distance set attaining the Fisher type upper bound, then determining a weight function $w$, $(X,w)$ is a tight weighted spherical $2k$-design. This result implies that locally $k$-distance sets attaining the Fisher type upper bound are $k$-distance sets. In the second part, we give a new absolute bound for the cardinalities of $k$-distance sets on a sphere. This upper bound is useful for $k$-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in $(d-1)$-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in $d$-space with more than $d(d+1)/2$ points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.Comment: 17 pages, 1 figur