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Note on the product of the largest and the smallest eigenvalue of a graph

Aida Abiad, Cristina Dalfó, Miquel Àngel Fiol

Abstract

In this note, we use eigenvalue interlacing to derive an inequality between the maximum degree of a graph and its maximum and minimum adjacency eigenvalues. The case of equality is fully characterized.

Note on the product of the largest and the smallest eigenvalue of a graph

Abstract

In this note, we use eigenvalue interlacing to derive an inequality between the maximum degree of a graph and its maximum and minimum adjacency eigenvalues. The case of equality is fully characterized.
Paper Structure (4 sections, 5 theorems, 19 equations, 1 table)

This paper contains 4 sections, 5 theorems, 19 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The new bound
  4. Bounds comparison

Key Result

Theorem 1

Let $\hbox{\boldmath $S$}$ be a real $n \times m$ matrix such that $\hbox{\boldmath $S$}^T \hbox{\boldmath $S$} = \hbox{\boldmath $I$}$, and let $\hbox{\boldmath $A$}$ be an $n \times n$ matrix with eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$. Define $\hbox{\boldmath $B$} = \hbox{\boldmath $S

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • Theorem 3
  • proof
  • Corollary 4
  • Proposition 5: adf2013