Let G be a connected graph on n vertices with weights assigned to each of its edges. If <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R=(r_{ij})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the resistance matrix of G, then the resistance Laplacian of G is <Equation ID="Equ17"> <EquationSource Format="TEX">$\begin{aligned} {\widetilde{R}}:=\textrm{Diag}\left( \sum _{j=1}^nr_{1j},\dotsc ,\sum _{j=1}^{n} r_{nj}\right) -R. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover accent="true"> <mi>R</mi> <mo stretchy="false">~</mo> </mover> <mo>:</mo> <mo>=</mo> <mtext>Diag</mtext> <mfenced close=")" open="("> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>r</mi> <mrow> <mi mathvariant="italic">nj</mi> </mrow> </msub> </mfenced> <mo>-</mo> <mi>R</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(x_1,\dotsc ,x_n)'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> be an eigenvector corresponding to the largest eigenvalue of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\widetilde{R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>R</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>. In this paper, we establish that the subgraphs induced by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{j: x_j \ge 0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>j</mi> <mo>:</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\{j:x_j \le 0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>j</mi> <mo>:</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>≤</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> are connected components of G.

Spectral partition of connected graphs by resistance Laplacian matrices

期刊论文
发表日期：

2026年4月7日

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Vinayak Gupta,
Gargi Lather,
R. Balaji

摘要

Let G be a connected graph on n vertices with weights assigned to each of its edges. If $R=(r_{ij})$ $R = (r_{ij})$ is the resistance matrix of G, then the resistance Laplacian of G is $\begin{aligned} {\widetilde{R}}:=\textrm{Diag}\left( \sum _{j=1}^nr_{1j},\dotsc ,\sum _{j=1}^{n} r_{nj}\right) -R. \end{aligned}$ $\begin{matrix} \tilde{R} : = Diag (\sum_{j = 1}^{n} r_{1 j}, \dots, \sum_{j = 1}^{n} r_{nj}) - R . \end{matrix}$ Let $(x_1,\dotsc ,x_n)'$ ${(x_{1}, \dots, x_{n})}^{'}$ be an eigenvector corresponding to the largest eigenvalue of ${\widetilde{R}}$ $\tilde{R}$ . In this paper, we establish that the subgraphs induced by $\{j: x_j \ge 0\}$ ${j : x_{j} \geq 0}$ and $\{j:x_j \le 0\}$ ${j : x_{j} \leq 0}$ are connected components of G.