On Quadratic Embedding Constants of Star Product Graphs

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 $\mathrm{Q}\mathrm{E}\mathrm{C}(G)$ is non-positive. For a finite star product of (finite or infinite) graphs $G=G_1\star \cdots \star G_r$ an estimate of $\mathrm{Q}\mathrm{E}\mathrm{C}(G)$ is obtained after a detailed analysis of the minimal solution of a certain algebraic equation. For the path graph $P_n$ an implicit formula for $\mathrm{Q}\mathrm{E}\mathrm{C}(P_n)$ is derived, and by limit argument $\mathrm{Q}\mathrm{E}\mathrm{C}(\mathbb{Z})=\mathrm{Q}\mathrm{E}\mathrm{C}({\mathbb{Z}}_{+})=-1\mathrm{/}2$ is shown. During the discussion a new integer sequence is found.