A Bijection between the d-dimensional simplices with all distances in {1,2} and the partitions of d+1 16/02/ Sascha Kurz

16/02/ Sascha Kurz An integral point set is a set of n points in the euclidean E d with integral distances between the points. We use the term simplex for a set of d+1 points in the euclidean E d such that not all the points are contained in a hyperplane.

16/02/ Sascha Kurz Similar to integral point sets we define integral simplices as simplices with integral distances between its points. The largest distance of a point set is called its diameter.

16/02/ Sascha Kurz Number of integral simplices by diameter and dimension.

16/02/ Sascha Kurz A partition of an integer n is an r-tuple of integers (i 1,...,i r ) with and There is clearly an unique integral simplex with diameter 1 in any dimension. By testing the sequence of the numbers of simplices with diameter 2 with N.J.A. Sloane's marvellous „Online-Encyclopedia of Integer Sequences“ we learned that it is one less than the sequence of partions.

16/02/ Sascha Kurz Theorem: The number of integral d-dimensional simplices with diameter at most 2 is the number of partitions of d+1.

16/02/ Sascha Kurz An integer triangle with its distance matrix.

16/02/ Sascha Kurz The bijection between partitions and integral simplices Partition: i=(4,3,3,2,1) Distance matrix D:

16/02/ Sascha Kurz We consider only the part above the main diagonal. The matrix contains strings of bold printed 1's. The length of such a block of 1's is one less than the corresponding summand of the partition. i=( )

16/02/ Sascha Kurz i=( ) Each such block of 1's is completed to an upper triangular matrix at the bottom of the corresponding columns.

16/02/ Sascha Kurz Proof of the bijection Every partition yields an integral simplex. Two different partitions yield two nonisomorphic integral simplices. For every integral simplex there is a corresponding partition.

16/02/ Sascha Kurz Every partition yields an integral simplex Not every symmetric matrix can be realized as a distance matrix in the euclidean space. There is, for example, no triangle with side lengths 4, 2, and 1. Definition: For a matrix A we define

16/02/ Sascha Kurz Theorem (Menger): If M is a set of d+1 points with distance maxtrix D=(d i,j ) and A=(d i,j 2 ), then M is realizeable in the euclidean d-dimensional space, iff and each subset of M is realizeable in the (d-1)- dimensional space.

16/02/ Sascha Kurz Two different partitions yield two nonisomorphic integral simplices. In general there are different distance matrices which describe the same integral simplex. So we define an unique representant for the set of these matrices. Now we only have to show that two different partitions yield two different representants of the corresponding matrices.

16/02/ Sascha Kurz For every integral simplex there is a corresponding partition For a proof we only need the triangle inequallity and the properties of the unique representant of a distance matrix.

16/02/ Sascha Kurz Generalizations Integral simplices with all distances in {1,2,3}. Simplices with at most two different side lengths  and . Maximum number of such nonisomorphic simplices  number of graphs Minimum number of nonisomorphic simplices. Description of graph classes corresponding to the simplices for special side lenghts  and   threshold functions