Further Results on the Quadratic Embedding Constants of Corona Graphs

The quadratic embedding constant (QEC) is a numerical invariant associated with quadratic embeddings of graphs into Hilbert spaces, and it is characterized in terms of the distance matrix. For corona graphs $G\odot H$, a general expression for $\mathrm{Q}\mathrm{E}\mathrm{C}(G\odot H)$ can be described using $\mathrm{Q}\mathrm{E}\mathrm{C}(G)$ together with spectral properties of $H$. However, this expression involves an additional spectral contribution determined by the adjacency matrix of $H$. In this paper, we analyze this contribution and provide an explicit description of the associated set $\mathrm{\Gamma }$, allowing us to determine the quantity $\gamma =\max \mathrm{\Gamma }$ that appears in the general formula for $\mathrm{Q}\mathrm{E}\mathrm{C}(G\odot H)$. As applications, we compute the quadratic embedding constants for corona graphs of the form $G\odot H$ where $H$ is a regular graph. Finally, we provide conditions on $G$ and $H$ under which the quadratic embedding constant of $G\odot H$ coincides with the second largest eigenvalue of the distance matrix.