On quadratic embeddability of bipartite graphs and theta graphs

TL;DR

This paper investigates the quadratic embedding properties of bipartite and theta graphs, calculating specific constants and identifying graphs that lack quadratic embeddability, thus advancing understanding of graph embedding limitations.

Contribution

It introduces new calculations of quadratic embedding constants for specific graph classes and identifies an infinite family of graphs that are not quadratically embeddable.

Findings

Computed quadratic embedding constants for certain bipartite graphs

Analyzed quadratic embedding properties of theta graphs

Identified an infinite family of non-quadratically embeddable graphs

Abstract

We compute the quadratic embedding constant for complete bipartite graphs with disjoint edges removed. Moreover, we study the quadratic embedding property for theta graphs, i.e., graphs consisting of three paths with common initial points and common endpoints. As a result, we provide an infinite family of primary graphs which are not quadratically embeddable.