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Weighted sums of rooted spanning forests on cycles with pendant edges

Hajime Fujita, Kimiko Hasegawa, Yukie Inaba, Takefumi Kondo

Abstract

We derive two formulas for the weighted sums of rooted spanning forests of particular sequence of graphs by using the matrix tree theorem. We consider cycle graphs with edges so called the pendant edges. One of our formula can be described as a variable transformation of the Chebyshev polynomial. They have particular algebraic properties.

Weighted sums of rooted spanning forests on cycles with pendant edges

Abstract

We derive two formulas for the weighted sums of rooted spanning forests of particular sequence of graphs by using the matrix tree theorem. We consider cycle graphs with edges so called the pendant edges. One of our formula can be described as a variable transformation of the Chebyshev polynomial. They have particular algebraic properties.
Paper Structure (10 sections, 11 theorems, 43 equations, 7 figures)

This paper contains 10 sections, 11 theorems, 43 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and terminologies
  4. Laplacian matrix and matrix tree theorem
  5. Non-oriented version
  6. Oriented version
  7. Main theorems : the weighted sum of cycles with $n$-pendant edges
  8. The non-oriented case
  9. The oriented case
  10. Corollaries and further discussions

Key Result

Theorem 2.4

If the node set consists of a single vertex, then we have where $L_{(v_iv_j)}(\mathcal{G})$ is the $(|V(\mathcal{G})|-1)\times(|V(\mathcal{G})|-1)$ matrix obtained by deleting a row vector corresponding to $i$-th vertex $v_i$ and a column vector corresponding to $j$-th vertex $v_j$ from the Laplacian matrix $L(\mathcal{G})$ of $\mathcal{G}$.

Figures (7)

  • Figure 1: $\mathcal{G}_5$
  • Figure 2: $\mathcal{G}_6$
  • Figure 3: $\tilde{\mathcal{G}}_5$
  • Figure 4: $\tilde{\mathcal{G}}_6$
  • Figure 5: an oriented rooted spanning forest
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: The Laplacian matrix
  • Theorem 2.4: The matrix tree theorem,Wagner
  • Definition 2.5: The oriented Laplacian matrix
  • Definition 2.6: The oriented rooted spanning forest
  • Definition 2.7: The weighted sum of oriented rooted spanning forest
  • Theorem 2.8: The oriented matrix tree theorem,TakasakiTutte
  • Definition 3.1
  • Remark 3.2
  • ...and 18 more