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Further Results on the Quadratic Embedding Constants of Corona Graphs

Ferdi, Edy Tri Baskoro, Aditya Purwa Santika

Abstract

The quadratic embedding constant (QEC) is a numerical invariant associated with quadratic embeddings of graphs into Hilbert spaces, and it is characterized in terms of the distance matrix. For corona graphs $G\odot H$, a general expression for $\mathrm{QEC}(G\odot H)$ can be described using $\mathrm{QEC}(G)$ together with spectral properties of $H$. However, this expression involves an additional spectral contribution determined by the adjacency matrix of $H$. In this paper, we analyze this contribution and provide an explicit description of the associated set $Γ$, allowing us to determine the quantity $γ= \max Γ$ that appears in the general formula for $\mathrm{QEC}(G\odot H)$. As applications, we compute the quadratic embedding constants for corona graphs of the form $G\odot H$ where $H$ is a regular graph. Finally, we provide conditions on $G$ and $H$ under which the quadratic embedding constant of $G\odot H$ coincides with the second largest eigenvalue of the distance matrix.

Further Results on the Quadratic Embedding Constants of Corona Graphs

Abstract

The quadratic embedding constant (QEC) is a numerical invariant associated with quadratic embeddings of graphs into Hilbert spaces, and it is characterized in terms of the distance matrix. For corona graphs , a general expression for can be described using together with spectral properties of . However, this expression involves an additional spectral contribution determined by the adjacency matrix of . In this paper, we analyze this contribution and provide an explicit description of the associated set , allowing us to determine the quantity that appears in the general formula for . As applications, we compute the quadratic embedding constants for corona graphs of the form where is a regular graph. Finally, we provide conditions on and under which the quadratic embedding constant of coincides with the second largest eigenvalue of the distance matrix.
Paper Structure (5 sections, 11 theorems, 165 equations)

This paper contains 5 sections, 11 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Quadratic Embedding Constants of Corona Graphs in General
  3. Explicit Description of the Corona Graph $G\odot H$
  4. Corona graph $G\odot H$ with $H$ being regular graph
  5. Quadratic Embedding Constants and the Second Distance Eigenvalue

Key Result

Theorem 1

Let $G=(V_1,E_1)$ be a connected graph with $|V_1|\ge 2$, and let $H=(V_2,E_2)$ be a graph with $|V_2|\ge 1$. Let $A_H$ denote the adjacency matrix of $H$ and define Assume that Then the quadratic embedding constant of the corona graph $G\odot H$ is given by where If the set $\Gamma$ is empty, we understand that $\gamma=-\infty$.

Theorems & Definitions (22)

  • Theorem 1: Ferdi-Baskoro-Obata-Santika2025
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • Proposition 6
  • ...and 12 more