Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programm
ing method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some dual technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected $k$-regular graph satisfying $g>2d-1$ has the minimum second-largest eigenvalue of all $k$-regular graphs of the same size, where $d$ is the number of distinct non-trivial eigenvalues, and $g$ is the girth. The known graphs satisfying $g>2d-1$ are Moore graphs, incidence graphs of regular generalized polygons of order $(s,s)$, triangle-free strongly regular graphs, and the odd graph of degree $4$.
In this paper we characterize large regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that large association schemes become $P$-polynomial association schemes. Ou
r results are summarized as follows. Let $G=(V,E)$ be a connected $k$-regular graph with $d+1$ distinct eigenvalues $k=theta_0>theta_1>cdots>theta_d$. Since the diameter of $G$ is at most $d$, we have the Moore bound [ |V| leq M(k,d)=1+k sum_{i=0}^{d-1}(k-1)^i. ] Note that if $|V|> M(k,d-1)$ holds, the diameter of $G$ is equal to $d$. Let $E_i$ be the orthogonal projection matrix onto the eigenspace corresponding to $theta_i$. Let $partial(u,v)$ be the path distance of $u,v in V$. Theorem. Assume $|V|> M(k,d-1)$ holds. Then for $x,y in V$ with $partial(x,y)=d$, the $(x,y)$-entry of $E_i$ is equal to [ -frac{1}{|V|}prod_{j=1,2,ldots,d, j e i} frac{theta_0-theta_j}{theta_i-theta_j}. ] If a symmetric association scheme $mathfrak{X}=(X,{R_i}_{i=0}^d)$ has a relation $R_i$ such that the graph $(X,R_i)$ satisfies the above condition, then $mathfrak{X}$ is $P$-polynomial. Moreover we show the dual version of this theorem for spherical sets and $Q$-polynomial association schemes.
Victoir (2004) developed a method to construct cubature formulae with various combinatorial objects. Motivated by this, we generalize Victoirs method with one more combinatorial object, called regular t-wise balanced designs. Many cubature of small i
ndices with few points are provided, which are used to update Shatalovs table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.
In this paper we consider the existence problem of cubature formulas of degree 4k+1 for spherically symmetric integrals for which the equality holds in the Moller lower bound. We prove that for sufficiently large dimensional minimal formulas, any two
distinct points on some concentric sphere have inner products all of which are rational numbers. By applying this result we prove that for any d > 2 there exist no d-dimensional minimal formulas of degrees 13 and 21 for some special integral.
In this paper, we simplify the known switching theorem due to Bose and Shrikhande as follows. Let $G=(V,E)$ be a primitive strongly regular graph with parameters $(v,k,lambda,mu)$. Let $S(G,H)$ be the graph from $G$ by switching with respect to a
nonempty $Hsubset V$. Suppose $v=2(k-theta_1)$ where $theta_1$ is the nontrivial positive eigenvalue of the $(0,1)$ adjacency matrix of $G$. This strongly regular graph is associated with a regular two-graph. Then, $S(G,H)$ is a strongly regular graph with the same parameters if and only if the subgraph induced by $H$ is $k-frac{v-h}{2}$ regular. Moreover, $S(G,H)$ is a strognly regualr graph with the other parameters if and only if the subgraph induced by $H$ is $k-mu$ regular and the size of $H$ is $v/2$. We prove these theorems with the view point of the geometrical theory of the finite set on the Euclidean unit sphere.
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 a
re 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.
