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Fast Consensus Topology Design via Minimizing Laplacian Energy

Susie Lu, Ji Liu

Abstract

This paper characterizes the graphical properties of an optimal topology with minimal Laplacian energy under the constraint of fixed numbers of vertices and edges, and devises an algorithm to construct such connected optimal graphs. These constructed graphs possess maximum vertex and edge connectivity, and more importantly, exhibit large algebraic connectivity of an optimal order provided they are not sparse. These properties guarantee fast and resilient consensus processes over these graphs.

Fast Consensus Topology Design via Minimizing Laplacian Energy

Abstract

This paper characterizes the graphical properties of an optimal topology with minimal Laplacian energy under the constraint of fixed numbers of vertices and edges, and devises an algorithm to construct such connected optimal graphs. These constructed graphs possess maximum vertex and edge connectivity, and more importantly, exhibit large algebraic connectivity of an optimal order provided they are not sparse. These properties guarantee fast and resilient consensus processes over these graphs.
Paper Structure (5 sections, 14 theorems, 17 equations, 13 figures)

This paper contains 5 sections, 14 theorems, 17 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Minimal Laplacian Energy
  3. Connectivity Resilience
  4. Fast Consensus
  5. Conclusion

Key Result

Theorem 1

Among all simple graphs with $n$ vertices and $m$ edges, the minimal Laplacian energy is $(k+1)(4m-nk)$ with $k = \lfloor \frac{2m}{n} \rfloor$, which is achieved if, and only if, $n(k+1)-2m$ vertices are of degree $k$ and the remaining $2m-nk$ vertices are of degree $k+1$.

Figures (13)

  • Figure 1: Maximal algebraic connectivity graphs with 4 vertices
  • Figure 2: Maximal algebraic connectivity graphs with 5 vertices
  • Figure 3: Maximal algebraic connectivity graphs with 6 vertices
  • Figure 4: Maximal algebraic connectivity graphs with 7 vertices
  • Figure 5: All minimal Laplacian energy graphs with 6 vertices and 6 edges
  • ...and 8 more figures

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 4
  • Lemma 3
  • Theorem 5
  • ...and 5 more