Quadratic embedding constants of graphs: Bounds and distance spectra

TL;DR

This paper investigates the quadratic embedding constants (QEC) of graphs, establishing bounds for trees, exploring their relation to distance spectra, and identifying new classes of graphs with specific QEC properties.

Contribution

It provides sharp bounds for QEC of trees, links QEC to distance spectra, and introduces new graph classes with computable QEC, advancing understanding of graph embeddings.

Findings

QEC of a graph is zero iff its second largest distance eigenvalue is zero.

Identified a subclass of nonsingular graphs where QEC equals the second largest distance eigenvalue.

Derived formulas for QEC of graph clusters involving complete or star graphs.

Abstract

The quadratic embedding constant (QEC) of a finite, simple, connected graph $G$ is the maximum of the quadratic form of the distance matrix of $G$ on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs was motivated by the classical work of Schoenberg on quadratic embedding of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In this article, we provide sharp upper and lower bounds for the QEC of trees. We next explore the relation between distance spectra and quadratic embedding constants of graphs - and show two further results: $(i)$ We show that the quadratic embedding constant of a graph is zero if and only if its second largest distance eigenvalue is zero. $(ii)$ We identify a new subclass of nonsingular graphs whose QEC is the second largest distance eigenvalue. Finally, we show that the QEC of the cluster of an arbitrary graph $G$ with either a complete or star graph can be computed in terms of the QEC of $G$. As an application of this result, we provide new families of examples of graphs of QE class.