Further Results on the Quadratic Embedding Constants of Corona Graphs

TL;DR

This paper investigates the quadratic embedding constants of corona graphs, providing explicit formulas and conditions for their computation, especially when the second largest eigenvalue of the distance matrix coincides with the QEC.

Contribution

It offers a new explicit description of the spectral contribution to QEC of corona graphs and computes QEC for specific regular graphs, advancing understanding of graph embeddings.

Findings

Explicit formula for QEC of corona graphs involving spectral properties.

Computed QEC for corona graphs where H is a regular graph.

Identified conditions where QEC equals the second largest eigenvalue of the distance matrix.

Abstract

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 $\Gamma$, allowing us to determine the quantity $\gamma = \max \Gamma$ 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.