Quadratic Embedding Constants of Cartesian Products and Joins of Graphs

TL;DR

This paper investigates the quadratic embedding constant (QEC) of graphs, focusing on how it behaves under Cartesian product and join operations, providing formulas, bounds, and new examples.

Contribution

It derives general formulas for the QEC of graph joins with regular and multipartite graphs, and establishes bounds for Cartesian products, advancing understanding of graph embedding properties.

Findings

Explicit formulas for QEC of joins with regular and multipartite graphs

Lower bounds for QEC of Cartesian products of connected graphs

New examples of graphs belonging to the QE class

Abstract

The quadratic embedding constant (QEC) of a finite, simple, connected graph originated from the classical work of Schoenberg [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938] on Euclidean distance geometry. In this article, we study the QEC of graphs in terms of two graph operations: the Cartesian product and the join of graphs. We derive a general formula for the QEC of the join of an arbitrary graph with a regular graph and with a complete multipartite graph. As an application of these results, we explicitly compute the QEC for several classes of graphs and provide new examples of graphs of QE class. We also establish a lower bound for the quadratic embedding constant of the Cartesian product of two arbitrary connected graphs. Furthermore, as an extremal case, we derive concise formulas for the quadratic embedding constants of the Cartesian product of an arbitrary graph G with a complete graph and with a complete bipartite graph, expressed in terms of $\qec(G)$.