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Solution to an open problem on Laplacian ratio

T. Wu

Abstract

Let G be a graph. The Laplacian ratio of G is the permanent of the Laplacian matrix of G divided by the product of degrees of all vertices. The computational complexity of Laplacian ratio is #P-complete. Brualdi and Goldwasser studied systematicly the properties of Laplacian ratios of graphs. And they proposed an open problem: what is the minimum value of the Laplacian ratios of trees with n vertices having diameter at least k ? In this paper, we give a solution to the problem.

Solution to an open problem on Laplacian ratio

Abstract

Let G be a graph. The Laplacian ratio of G is the permanent of the Laplacian matrix of G divided by the product of degrees of all vertices. The computational complexity of Laplacian ratio is #P-complete. Brualdi and Goldwasser studied systematicly the properties of Laplacian ratios of graphs. And they proposed an open problem: what is the minimum value of the Laplacian ratios of trees with n vertices having diameter at least k ? In this paper, we give a solution to the problem.
Paper Structure (3 sections, 12 theorems, 56 equations, 4 figures)

This paper contains 3 sections, 12 theorems, 56 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Summarize

Key Result

Theorem 1.1

Let $T$ be a tree with $n$ vertices and diameter at least $k$, Then where equality holds if and only if tree $T$ is the broom graph $B(n,k)$.

Figures (4)

  • Figure 1: The Broom graph $B(n,k)$.
  • Figure 2: Graphs $G_{1}$ and $G_{2}$.
  • Figure 3: Graphs $G_{1}$ and $G_{1}^{'}$.
  • Figure 4: Graphs $T$, $T_{1}$ and $T_{2}$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 8 more