<b>Locating Eigenvalues of Perturbed Laplacian Matrices of Trees</b>

We give a linear time algorithm to compute the number of eigenvalues of any perturbed Laplacian matrix of a tree in a given real interval. The algorithm can be applied to weighted or unweighted trees. Using our method we characterize the trees that have up to 5 distinct eigenvalues with respect to a family of perturbed Laplacian matrices that includes the adjacency and normalized Laplacian matrices as special cases, among others.

Keywords:
perturbed Laplacian matrix; eigenvalue location; trees

Authorship

R.O. BRAGA ^**Corresponding author: Rodrigo Orsini Braga - E-mail: rbraga@ufrgs.br

Departamento de Matemática Pura e Aplicada, IME - UFRGS, Av. Bento Gonçalves, 9500, 91509-900 Porto Alegre, RS, Brasil.UFRGSBrazilPorto Alegre, RS, BrazilDepartamento de Matemática Pura e Aplicada, IME - UFRGS, Av. Bento Gonçalves, 9500, 91509-900 Porto Alegre, RS, Brasil.

1 This work is part of the doctoral studies of this author.

V.M. RODRIGUES

Departamento de Matemática Pura e Aplicada, IME - UFRGS, Av. Bento Gonçalves, 9500, 91509-900 Porto Alegre, RS, Brasil. E-mail: vrodrig@mat.ufrgs.brUFRGSBrazilPorto Alegre, RS, BrazilDepartamento de Matemática Pura e Aplicada, IME - UFRGS, Av. Bento Gonçalves, 9500, 91509-900 Porto Alegre, RS, Brasil. E-mail: vrodrig@mat.ufrgs.br

*Corresponding author: Rodrigo Orsini Braga - E-mail: rbraga@ufrgs.br

1
This work is part of the doctoral studies of this author.

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (6)
Formulas (24)

Thumbnail

Algorithm 1
DiagonalizeW(T, α).

Thumbnail

Figure 1:
Algorithm DiagonalizeW(T, α) with α = −2.

Thumbnail

Figure 2:
A diameter 3 tree.

Thumbnail

Figure 3:
A diameter 4 tree T (k, p₁, p₂, ..., p_k, m).

Thumbnail

Figure 4:
A diameter 4 tree with m = 0.

Thumbnail

Figure 5:
A diameter 4 tree with m > 0.

Thumbnail

Algorithm 1 DiagonalizeW(T, α).

Figure 1: Algorithm DiagonalizeW(T, α) with α = −2.

Figure 2: A diameter 3 tree.

Figure 3: A diameter 4 tree T (k, p₁, p₂, ..., p_k, m).

Figure 4: A diameter 4 tree with m = 0.

Figure 5: A diameter 4 tree with m > 0.

{}_{D}L (G) = D - A,

\begin{matrix} k \\ j \end{matrix} [\begin{matrix} ⋮ & ⋮ \\ \dots & a (v_{k}) & \dots & ω_{k j} & \dots \\ ⋮ & ⋱ & ⋮ \\ \dots & ω_{j k} & \dots & a (v_{j}) & \dots \\ ⋮ & ⋮ \end{matrix}] .

R_{k} \leftarrow R_{k} - \frac{ω_{j k}}{a (υ_{j})} R_{j} and C_{k} \leftarrow C_{k} - \frac{ω_{j k}}{a (υ_{j})} C_{j} .

\begin{matrix} k \\ j \end{matrix} [\begin{matrix} ⋮ & ⋮ \\ \dots & a (v_{k}) - \frac{(ω_{k j})^{2}}{a (v_{j})} & \dots & 0 & \dots \\ ⋮ & ⋱ & ⋮ \\ \dots & 0 & \dots & a (v_{j}) & \dots \\ ⋮ & ⋮ \end{matrix}] .

a (v_{k}) - \sum_{v_{i} \in C_{k}} \frac{(ω_{i k})^{2}}{a (v_{i})},

\begin{matrix} ℓ \\ k \\ j \\ i \end{matrix} [\begin{matrix} δ_{ℓ} + α & ω_{ℓ k} \\ ω_{k ℓ} & δ_{k} + α & ω_{k j} & ω_{k i} \\ ω_{j k} & 0 \\ ω_{i k} & a (v_{i}) \end{matrix}] .

R_{i} \leftarrow R_{i} - \frac{ω_{i k}}{ω_{j k}} R_{j} and C_{i} \leftarrow C_{i} - \frac{ω_{k i}}{ω_{k j}} C_{j},

R_{ℓ} \leftarrow R_{ℓ} - \frac{ω_{ℓ k}}{ω_{j k}} R_{j} and C_{ℓ} \leftarrow C_{ℓ} - \frac{ω_{k ℓ}}{ω_{k j}} C_{j} .

\begin{matrix} ℓ \\ k \\ j \\ i \end{matrix} [\begin{matrix} δ_{ℓ} + α & 0 \\ 0 & δ_{k} + α & ω_{k j} & 0 \\ ω_{j k} & 0 \\ 0 & a (v_{i}) \end{matrix}] .

R_{k} \leftarrow R_{k} - \frac{(δ_{k} + α)}{2 ω_{j k}} R_{j} and C_{k} \leftarrow C_{k} - \frac{(δ_{k} + α)}{2 ω_{k j}} C_{j}

\begin{matrix} ℓ \\ k \\ j \\ i \end{matrix} [\begin{matrix} δ_{ℓ} + α & 0 \\ 0 & 0 & ω_{k j} & 0 \\ ω_{j k} & 0 \\ 0 & a (v_{i}) \end{matrix}] .

R_{j} \leftarrow R_{j} + \frac{1}{ω_{k j}} R_{k}, C_{j} \leftarrow C_{j} + \frac{1}{ω_{j k}} C_{k}, R_{k} \leftarrow R_{k} - \frac{ω_{k j}}{2} R_{j}, C_{k} \leftarrow C_{k} - \frac{ω_{j k}}{2} C_{j}

\begin{matrix} ℓ \\ k \\ j \\ i \end{matrix} [\begin{matrix} δ_{ℓ} + α & 0 \\ 0 & - \frac{(ω_{k j})^{2}}{2} & 0 & 0 \\ 0 & 2 \\ 0 & a (v_{i}) \end{matrix}] .

a (v_{3}) = - 1 - \frac{(ω_{34})^{2}}{a (v_{4})} - \frac{(ω_{35})^{2}}{a (v_{5})} = - 1 - \frac{2^{2}}{- 1} - \frac{1^{2}}{- 1} = 4 .

a (v_{1}) = - \frac{(ω_{12})^{2}}{2} = - \frac{1^{2}}{2} = - \frac{1}{2} .

\sum_{i = 1}^{k} d_{i} ⩽ \sum_{i = 1}^{k} λ_{i}, for k = 1, \dots, n - 1, and \sum_{i = 1}^{n} d_{i} = \sum_{i = 1}^{n} λ_{i} .

m = 1 + \max_{1 ⩽ k ⩽ n} \{λ_{k}\},

{}_{D}L (G) = μ I - A .

ω_{i j} = \frac{1}{\sqrt{d_{i} d_{j}}},

\sum_{i = 1}^{k} (p_{i} - 1) + 1 = (\sum_{i = 1}^{k} p_{i}) - k + 1 ⩾ 1 .

(\sum_{i = 1}^{k} p_{i}) - k + (m - 1) . □

a (v_{i}) = μ - λ - \sum_{h = 1}^{p_{i}} \frac{(ω_{i h})^{2}}{μ - λ} = μ - λ - \frac{σ_{i}}{μ - λ},

a (v_{i}) = a (v_{j}) \Leftrightarrow μ - λ - \frac{σ_{i}}{μ - λ} = μ - λ - \frac{σ_{j}}{μ - λ} \Leftrightarrow σ_{i} = σ_{j} .

a (v_{i}) = μ - λ - \frac{σ_{i}}{μ - λ} = \frac{(μ - λ)^{2} - σ}{μ - λ} = 0 .

How to cite

copy

Sociedade Brasileira de Matemática Aplicada e Computacional Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163 , 13561-120 São Carlos - SP, Tel. / Fax: (55 16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br

Acompanhe os números deste periódico no seu leitor de RSS

Versão para download de PDF

PDF

Inglês

Versões e tradução automática

Versão original do texto

English

Tradução automática

Google Translator
Microsoft Translator

Como citar

RIS

BIBTEX

Outros formatos de citação e exportação:

SciELO - Scientific Electronic Library Online
Rua Dr. Diogo de Faria, 1087 – 9º andar – Vila Clementino 04037-003 São Paulo/SP - Brasil
E-mail: scielo@scielo.org