A Classification of Graphs through Quadratic Embedding Constants and Clique Graph Insights

TL;DR

This paper introduces the quadratic embedding constant (QEC) as a new graph invariant, explores its relation to graph structure via clique graphs, and characterizes graphs based on their QEC values, including specific classes like paths and cactus-like structures.

Contribution

It defines the quadratic embedding constant, derives formulas for certain graph classes, and characterizes graphs with QEC below specific thresholds, advancing understanding of graph embeddings.

Findings

Graphs with QEC < -1/2 have a cactus-like structure.

Derived a formula for QEC of graphs with two maximal cliques.

Characterized graphs with QEC less than that of P_5.

Abstract

The quadratic embedding constant (QEC) of a graph $G$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $\mathrm{QEC}(G)$. By observing graph structure of the maximal cliques (clique graph), we show that a graph $G$ with $\mathrm{QEC}(G)<-1/2$ admits a ``cactus-like'' structure. We derive a formula for the quadratic embedding constant of a graph consisting of two maximal cliques. As an application we discuss characterization of graphs along the increasing sequence of $\mathrm{QEC}(P_d)$, where $P_d$ is the path on $d$ vertices. In particular, we determine graphs $G$ satisfying $\mathrm{QEC}(G)<\mathrm{QEC}(P_5)$.