It is proved that a graph on n nodes is k-connected if and only if its nodes can be represented by real vectors in dimension n - k such that (a) nonadjacent nodes are represented by orthogonal vectors and (b) any n - k of them are linearly independent. We show that the closure of the set of all representations with properties (a) and (b) is irreducible as an algebraic variety, and study the question of irreducibility of the variety of all representations with property (a).
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics