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 $\mathrm{QEC}(G\odot H)$ can be described using $\mathrm{QEC}(G)$ 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 $Γ$, allowing us to determine the quantity $γ= \max Γ$ that appears in the general formula for $\mathrm{QEC}(G\odot H)$. 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.

💡 Research Summary

The paper investigates the quadratic embedding constant (QEC) of corona graphs, a graph operation denoted by (G\odot H). The QEC measures whether a graph can be embedded into a Hilbert space preserving pairwise distances, and it is defined as the maximum of (\langle f, D_G f\rangle) over unit vectors orthogonal to the all‑ones vector, where (D_G) is the distance matrix. Earlier work gave a formula for (QEC(G\odot H)) that involved the term (\psi_H^{-1}(QEC(G))) together with an additional spectral contribution coming from a set (\Gamma). However, the role of (\Gamma) was not fully clarified, leaving the general expression incomplete.

The authors first recall the construction of the corona graph and the block structure of its distance matrix. By reformulating the QEC problem as a constrained eigenvalue problem (\max \lambda(S)), they analyze the set of admissible eigenvalues (\lambda(S)). They partition (\lambda(S)) into three subsets:

• (\Gamma_1) consists of values (\lambda) for which (-2-\lambda) is not an eigenvalue of the adjacency matrix (A_H) and (\lambda\neq-2). In this case (\lambda) coincides with the unique solution of (\psi_H(\lambda)=QEC(G)), i.e., (\lambda=\psi_H^{-1}(QEC(G))).
• (\Gamma_2) contains the single value (\lambda=-2). This value belongs to (\lambda(S)) only when either (QEC(G)=0) or (-2) itself is an eigenvalue of (A_H). Consequently the associated contribution (\gamma_2) is either (0) (under those special circumstances) or (-\infty) otherwise.
• (\Gamma_3) comprises values for which (-2-\lambda) is an eigenvalue of (A_H) different from (-2). The authors prove that (\lambda\in\lambda(S)) precisely when there exists a non‑zero vector (x) orthogonal to the all‑ones vector such that ((A_H+2+\lambda)x=0). This leads to the definition \