A connected graph G is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant QEC(G) is non-positive. For a finite star product of (finite or infinite) graphs G = G1 *...* Gr an estimate of QEC(G) is obtained after a detailed analysis of the minimal solution of a certain algebraic equation. For the path graph Pn an implicit formula for QEC(Pn) is derived, and by limit argument QEC(Z) = QEC(Z+) =-1=2 is shown. During the discussion a new integer sequence is found.
ASJC Scopus subject areas
- 数学 (全般)