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Register a new userBounds on point configurations determined by distances and dot products
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We study a family of variants of ErdH os unit distance problem, concerning distances and dot products between pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, we look for bounds on how many subsets of $k$ points satisfy a set of relationships between point pairs based on distances or dot products. We survey some of the recent work in the area and present several new, more general families of bounds.
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If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} in X$, a geodesic triangle $T={x_{1},x_{2},x_{3}}$ is the union of the three geodesics $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $delta$-hyperbolic in the Gromov sense if any side of $T$ is contained in a $delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $delta(X) =inf { deltageq 0:{0.3cm}$ X ${0.2cm}$ $text{is} {0.2cm} delta text{-hyperbolic} }.$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolycity constant in terms of some parameters of the graph. Denote by $mathcal{G}(n,m)$ the set of graphs $G$ with $n$ vertices and $m$ edges, and such that every edge has length $1$. In this work we estimate $A(n,m):=min{delta(G)mid G in mathcal{G}(n,m) }$ and $B(n,m):=max{delta(G)mid G in mathcal{G}(n,m) }$. In particular, we obtain good bounds for $B(n,m)$, and we compute the precise value of $A(n,m)$ for all values of $n$ and $m$. Besides, we apply these results to random graphs.
A Cartesian decomposition of a coherent configuration $cal X$ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of $cal X$ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration $cal X$ is thick, then there is a unique maximal Cartesian decomposition of $cal X$, i.e., there is exactly one internal tensor decomposition of $cal X$ into indecomposable components. In particular, this implies an analog of the Krull--Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.
The convex grabbing game is a game where two players, Alice and Bob, alternate taking extremal points from the convex hull of a point set on the plane. Rational weights are given to the points. The goal of each player is to maximize the total weight over all points that they obtain. We restrict the setting to the case of binary weights. We show a construction of an arbitrarily large odd-sized point set that allows Bob to obtain almost 3/4 of the total weight. This construction answers a question asked by Matsumoto, Nakamigawa, and Sakuma in [Graphs and Combinatorics, 36/1 (2020)]. We also present an arbitrarily large even-sized point set where Bob can obtain the entirety of the total weight. Finally, we discuss conjectures about optimum moves in the convex grabbing game for both players in general.
We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and asymptotic formulas describing a random realizable configuration, obtained either by sampling the points uniformly at random on the circle or by sampling a realizable configuration uniformly at random.
Let $X$ be a finite set in a complex sphere of $d$ dimension. Let $D(X)$ be the set of usual inner products of two distinct vectors in $X$. A set $X$ is called a complex spherical $s$-code if the cardinality of $D(X)$ is $s$ and $D(X)$ contains an imaginary number. We would like to classify the largest possible $s$-codes for given dimension $d$. In this paper, we consider the problem for the case $s=3$. Roy and Suda (2014) gave a certain upper bound for the cardinalities of $3$-codes. A $3$-code $X$ is said to be tight if $X$ attains the bound. We show that there exists no tight $3$-code except for dimensions $1$, $2$. Moreover we make an algorithm to classify the largest $3$-codes by considering representations of oriented graphs. By this algorithm, the largest $3$-codes are classified for dimensions $1$, $2$, $3$ with a current computer.
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