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Quadratic Embedding of Theta Graphs via Reproducing Kernel Hilbert Spaces

Marek Skrzypczyk

Abstract

The quadratic embedding property of graphs consisting of three paths (theta graphs) is fully characterised. For this aim, a theorem by Winkler (1985) is utilized. An alternative proof of that result using the RKHS technique is presented.

Quadratic Embedding of Theta Graphs via Reproducing Kernel Hilbert Spaces

Abstract

The quadratic embedding property of graphs consisting of three paths (theta graphs) is fully characterised. For this aim, a theorem by Winkler (1985) is utilized. An alternative proof of that result using the RKHS technique is presented.
Paper Structure (5 sections, 5 theorems, 68 equations, 9 figures)

This paper contains 5 sections, 5 theorems, 68 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Winkler's Theorem
  4. Quadratic Embedding of $\Theta(1,\beta,\gamma)$
  5. Quadratic Embedding of $\Theta(2,3,\gamma)$

Key Result

Theorem 1

Assume that $1\le\alpha\le\beta\le\gamma$, $\beta\ge2$. Then $\Theta(\alpha,\beta,\gamma)$ is of QE class if and only if either

Figures (9)

  • Figure 1: The graph $\Theta(2,3,5)$
  • Figure 2: Oriented graph $\Theta(1,2k,2l)'$
  • Figure 3: Spanning tree of $\Theta(1,2k,2l)'$
  • Figure 4: Oriented graph $\Theta(1,2k,2l+1)'$
  • Figure 5: Spanning tree of $\Theta(1,2k,2l+1)'$
  • ...and 4 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2: Winkler Winklertree
  • proof : Proof of Theorem \ref{['tree']}
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • proof