Matrices and graphs in Euclidean geometry

Ela Matrices and Graphs in Euclidean Geometry

Some examples of the interplay between matrix theory, graph theory and n-dimensional Euclidean geometry are presented. In particular, qualitative properties of interior angles in simplices are completely characterized. For right simplices, a relationship between the tree of legs and the circumscribed Steiner ellipsoids is proved.

Spatial Analysis in curved spaces with Non-Euclidean Geometry

The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type an...

A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints

Let K n be the cone of positive semideenite n n matrices and let A be an aane subspace of the space of symmetric matrices such that the intersection K n \ A is non-empty and bounded. Suppose that n 3 and that codim A = ? r+2 2 for some 1 r n?2. Then there is a matrix X 2 K n \A such that rank X r. We give a short geometric proof of this result, use it to improve a bound on realizability of weig...

AUTOMORPHISM GROUPS OF SOME NON-TRANSITIVE GRAPHS

An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for ij, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. Balaban introduced some monster graphs and then Randic computed complexit...

On Euclidean distance matrices of graphs