Locating Eigenvalues of a Symmetric Matrix whose Graph is Unicyclic

We present a linear-time algorithm that computes in a given real interval the number of eigenvalues of any symmetric matrix whose underlying graph is unicyclic. The algorithm can be applied to vertexand/or edge-weighted or unweighted unicyclic graphs. We apply the algorithm to obtain some general results on the spectrum of a generalized sun graph for certain matrix representations which include the Laplacian, normalized Laplacian and signless Laplacian matrices.

Keywords:
symmetric matrix; eigenvalue location; unicyclic graph

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 - E-mail: rbraga@ufrgs.brUFRGSBrasilPorto Alegre, RS, BrasilDepartamento de Matemática Pura e Aplicada, IME UFRGS, Av. Bento Gonçalves, 9500, 91509-900 Porto Alegre, RS, Brasil - E-mail: rbraga@ufrgs.br

https://orcid.org/0000-0002-6156-2213

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

https://orcid.org/0000-0002-1964-3327

R.O. SILVA

Departamento de Matemática Pura e Aplicada, IME UFRGS, Av. Bento Gonçalves, 9500, 91509-900 Porto Alegre, RS, Brasil - E-mail: rafa.2604@hotmail.comUFRGSBrasilPorto Alegre, RS, BrasilDepartamento de Matemática Pura e Aplicada, IME UFRGS, Av. Bento Gonçalves, 9500, 91509-900 Porto Alegre, RS, Brasil - E-mail: rafa.2604@hotmail.com

https://orcid.org/0000-0001-6996-2142

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

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (11)
Tables (1)
Formulas (23)

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Algorithm 1
Algorithm DiagSMUnicyclic(G, α)

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Algorithm 2
DiagonalizeW(T, α) ³3 R.O. Braga & V.M. Rodrigues . Locating eigenvalues of perturbed Laplacian matrices of trees. TEMA Tend. Mat. Apl. Comput., 18(3) (2017), 479-491. doi:10.5540/tema.2017.018.03.479. URL https://doi.org/10.5540/tema.2017.018.03.479.
https://doi.org/10.5540/tema.2017.018.03...

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Algorithm 3
Procedure DiagSMCycle(a ₁ , a ₂ , ..., a _b )

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Figure 1:
Unicyclic graph G of Example 2.1.

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Figure 2:
Vertex-and edge-weighted graph G(L).

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Figure 3:
Step 1 and 2 of Algorithm 1 with

α = - 3

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Figure 4:
Initialization of procedure DiagSMCycle (Algorithm 3).

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Figure 5:
Iteration

i = 4

of procedure DiagSMCycle.

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Figure 6:
Iteration

i = 3

of procedure DiagSMCycle.

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Figure 7:
Final diagonal values for G(L) and

α = - 3

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Figure 8:
Balanced sun graph 𝒮_6,3 .

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Table 1:
The spectra of the 𝒮_b,p for matrices L, Q and ℒ

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Algorithm 1 Algorithm DiagSMUnicyclic(G, α)

Algorithm 2 DiagonalizeW(T, α) ³3 R.O. Braga & V.M. Rodrigues . Locating eigenvalues of perturbed Laplacian matrices of trees. TEMA Tend. Mat. Apl. Comput., 18(3) (2017), 479-491. doi:10.5540/tema.2017.018.03.479. URL https://doi.org/10.5540/tema.2017.018.03.479.
https://doi.org/10.5540/tema.2017.018.03...

Algorithm 3 Procedure DiagSMCycle(a ₁ , a ₂ , ..., a _b )

Figure 1: Unicyclic graph G of Example 2.1.

Figure 2: Vertex-and edge-weighted graph G(L).

Figure 3: Step 1 and 2 of Algorithm 1 with $α = - 3$ .

Figure 4: Initialization of procedure DiagSMCycle (Algorithm 3).

Figure 5: Iteration $i = 4$ of procedure DiagSMCycle.

Figure 6: Iteration $i = 3$ of procedure DiagSMCycle.

Figure 7: Final diagonal values for G(L) and $α = - 3$ .

Figure 8: Balanced sun graph 𝒮_6,3 .

Table 1: The spectra of the 𝒮_b,p for matrices L, Q and ℒ

Matrix	Spectrum
L	$\{1^{n - 2 b}, \frac{p + μ_{i} + 1}{2} \pm \sqrt{{(\frac{p + μ_{i} + 1}{2})}^{2} - μ_{i}}\}, where μ_{i} = 2 - 2 \cos (\frac{2 π i}{b})$
Q	$\{1^{n - 2 b}, \frac{p + μ_{i} + 1}{2} \pm \sqrt{{(\frac{p + μ_{i} + 1}{2})}^{2} - μ_{i}}\}, where μ_{i} = 2 + 2 \cos (\frac{2 π i}{b})$
ℒ	$\{1^{n - 2 b}, \frac{μ_{i} + 1}{2} \pm \sqrt{{(\frac{μ_{i} - 1}{2})}^{2} + \frac{p}{p + 2}}\}, where μ_{i} = 1 - \frac{2}{p + 2} \cos (\frac{2 π i}{b})$

[\begin{matrix} B & E_{1} & E_{2} & \dots & E_{b} \\ (E_{1})^{T} & B_{1} & 0 & \dots & 0 \\ (E_{2})^{T} & 0 & B_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ (E_{b})^{T} & 0 & 0 & \dots & B_{b} \end{matrix}],

[\begin{matrix} \tilde{B} & 0 & 0 & \dots & 0 \\ 0 & D_{1} & 0 & \dots & 0 \\ 0 & 0 & D_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & D_{b} \end{matrix}],

\begin{matrix} 1 \\ ⋮ \\ j - 1 \\ j \\ ⋮ \\ b \end{matrix} [\begin{matrix} a_{1} & \dots & z_{j - 1} & z_{j} & \dots & z_{b} \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ z_{j - 1} & \dots & a_{j - 1} & ω_{j, j - 1} & \dots & 0 \\ z_{j} & \dots & ω_{j, j - 1} & a_{j} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ z_{b} & \dots & 0 & 0 & \dots & a_{b} \end{matrix}] .

R_{j - 1} \leftarrow R_{j - 1} - \frac{ω_{j, j - 1}}{a_{j}} R_{j} C_{j - 1} \leftarrow C_{j - 1} - \frac{ω_{j, j - 1}}{a_{j}} C_{j} R_{1} \leftarrow R_{1} - \frac{z_{j}}{a_{j}} R_{j} C_{1} \leftarrow C_{1} - \frac{z_{j}}{a_{j}} C_{j}

\begin{matrix} 1 \\ ⋮ \\ j - 1 \\ j \\ ⋮ \\ b \end{matrix} [\begin{matrix} {\tilde{a}}_{1} & \dots & {\tilde{z}}_{j - 1} & 0 & \dots & 0 \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ {\tilde{z}}_{j - 1} & \dots & {\tilde{a}}_{j - 1} & 0 & \dots & 0 \\ 0 & \dots & 0 & a_{j} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & \dots & 0 & 0 & \dots & a_{b} \end{matrix}],

R_{j - 2} \leftarrow R_{j - 2} - \frac{ω_{j - 2, j - 1}}{ω_{j, j - 1}} R_{j} and C_{j - 2} \leftarrow C_{j - 2} - \frac{ω_{j - 2, j - 1}}{ω_{j, j - 1}} C_{j} .

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

\begin{matrix} 1 \\ ⋮ \\ j - 2 \\ j - 1 \\ j \\ ⋮ \\ b \end{matrix} [\begin{matrix} a_{1} & \dots & {\tilde{z}}_{j - 2} & β & γ & \dots & 0 \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ \\ {\tilde{z}}_{j - 2} & \dots & a_{j - 2} & 0 & 0 & \dots & 0 \\ β & \dots & 0 & - \frac{(ω_{j, j - 1})^{2}}{2} & 0 & \dots & 0 \\ γ & \dots & 0 & 0 & 2 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & \dots & 0 & 0 & 0 & \dots & a_{b} \end{matrix}],

β = \frac{1}{2} \cdot (z_{j - 1} - z_{j} \cdot (\frac{a_{j - 1}}{2 ω_{j, j - 1}} + ω_{j, j - 1}))

γ = z_{j} \cdot (1 - \frac{a_{j - 1}}{2 (ω_{j, j - 1})^{2}}) + \frac{z_{j - 1}}{ω_{j, j - 1}} .

R_{1} \leftarrow R_{1} + 2 β \frac{1}{(ω_{j, j - 1})^{2}} R_{j - 1} C_{1} \leftarrow C_{1} + 2 β \frac{1}{(ω_{j, j - 1})^{2}} C_{j - 1} R_{1} \leftarrow R_{1} - \frac{γ}{2} R_{j} C_{1} \leftarrow C_{1} - \frac{γ}{2} C_{j}

\begin{matrix} 1 \\ ⋮ \\ j - 2 \\ j - 1 \\ j \\ ⋮ \\ b \end{matrix} [\begin{matrix} {\tilde{a}}_{1} & \dots & {\tilde{z}}_{j - 2} & 0 & 0 & \dots & 0 \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ \\ {\tilde{z}}_{j - 2} & \dots & a_{j - 2} & 0 & 0 & \dots & 0 \\ 0 & \dots & 0 & - \frac{(ω_{j, j - 1})^{2}}{2} & 0 & \dots & 0 \\ 0 & \dots & 0 & 0 & 2 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & \dots & 0 & 0 & 0 & \dots & a_{b} \end{matrix}] .

β = \frac{1}{2} (z_{2} + z_{3} (1 + \frac{a_{2}}{2})) = \frac{1}{2} (- 1 + 1 (1 + \frac{7}{2} . \frac{1}{2})) = \frac{7}{8}; γ = z_{3} (1 - \frac{a_{2}}{2}) - z_{2} = 1 (1 - \frac{7}{2} . \frac{1}{2}) - (- 1) = \frac{1}{4}

a_{1} = \frac{3}{2} + 2 β^{2} - \frac{γ^{2}}{2} = \frac{3}{2} + 2 {(\frac{7}{8})}^{2} - \frac{1}{2} {(\frac{1}{4})}^{2} = 3 .

a (v_{i}) = p + 2 - λ - \frac{p}{1 - λ},

2 + p - λ - \frac{p}{1 - λ} - 2 = - μ \Leftrightarrow λ^{2} - (p + μ + 1) λ + μ = 0 \Leftrightarrow λ = \frac{p + μ + 1}{2} \pm \sqrt{{(\frac{p + μ + 1}{2})}^{2} - μ} .

[\begin{matrix} 1 & - \frac{1}{p + 2} & 0 & 0 & \dots & 0 & - \frac{1}{p + 2} \\ - \frac{1}{p + 2} & 1 & - \frac{1}{p + 2} & 0 & \dots & 0 & 0 \\ 0 & - \frac{1}{p + 2} & 1 & - \frac{1}{p + 2} & \dots & 0 & 0 \\ 0 & \dots & - \frac{1}{p + 2} & 1 & - \frac{1}{p + 2} & \dots & 0 \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & - \frac{1}{p + 2} \\ - \frac{1}{p + 2} & 0 & 0 & 0 & \dots & - \frac{1}{p + 2} & 1 \end{matrix}],

1 - λ - \frac{p}{(1 - λ) (p + 2)} - 1 = - μ \Leftrightarrow λ^{2} - (μ + 1) λ + μ - \frac{p}{p + 2} = 0 \Leftrightarrow λ = \frac{μ + 1}{2} \pm \frac{1}{2} \sqrt{(μ - 1)^{2} + \frac{4 p}{p + 2}} .

λ_{1} ⩽ \frac{p + 5 + \sqrt{(p + 1) (p + 9)}}{2},

θ_{1} = \frac{p + 5 + \sqrt{(p + 1) (p + 9)}}{2} .

λ_{1} = \frac{p + μ_{⌊\frac{b}{2}⌋} + 1}{2} + \sqrt{{(\frac{p + μ_{⌊\frac{b}{2}⌋} + 1}{2})}^{2} - μ_{⌊\frac{b}{2}⌋}} ⩽ \frac{p + 5}{2} + \sqrt{{(\frac{p + 5}{2})}^{2} - 4} = \frac{p + 5}{2} + \frac{1}{2} \sqrt{(p + 1) (p + 9)} .

a_{b, p} ⩽ \frac{(p + 4) - \sqrt{p^{2} + 8 p + 4}}{2}

a_{b, p} = \frac{p + μ_{b - 1} + 1}{2} - \sqrt{{(\frac{p + μ_{b - 1} + 1}{2})}^{2} - μ_{b - 1}} ⩽ \frac{p + 4}{2} - \sqrt{{(\frac{p + 4}{2})}^{2} - 3} = \frac{p + 4}{2} - \frac{1}{2} \sqrt{p^{2} + 8 p + 4} .

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Locating Eigenvalues of a Symmetric Matrix whose Graph is Unicyclic

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