Quadratic Embedding Constants of Corona Graphs

TL;DR

This paper derives a formula for the quadratic embedding constant of corona graphs, linking it to spectral properties of the component graphs and providing explicit formulas for certain regular graphs.

Contribution

The paper introduces a new formula for the QEC of corona graphs, connecting it to eigenvalues and inverse functions of an analytic function based on the component graph's spectrum.

Findings

Derived a formula for QEC of corona graphs

Established spectral conditions for the formula to hold

Provided explicit formulas for regular graphs with specific eigenvalues

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 $\mathrm{QEC}(G\odot H)=\psi_{H*}^{-1}(\mathrm{QEC}(G))$ holds under some spectral assumptions on $H$, where $\psi_{H*}^{-1}$ is the inverse function of the most right branch of the analytic function $\psi_H$ 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.