Quadratic Embedding Constants of Corona Graphs

Abstract

The quadratic embedding constant (QEC) of a connected graph is defined to be the maximum of the quadratic function associated with its distance matrix on a certain unit sphere of codimension two. In this paper we derive a formula for the QEC of a corona graph $G\odot H$. It is shown that $\mathrm{QEC}(G\odot H)=\ddot{I}{ˆ}_{H\ast }^{-1}(\mathrm{QEC}(G))$ holds under some spectral assumptions on $H$, where $\ddot{I}{ˆ}_{H\ast }^{-1}$ is the inverse function of the most right branch of the analytic function $\ddot{I}{ˆ}_H$ defined by means of the main eigenvalues of the adjacency matrix of $H$. Moreover, if $H$ is a regular graph of which the adjacency matrix has the smallest eigenvalue $-2$, then the formula is written down explicitly.