Let G be a connected graph with n vertices, m edges and having distance Laplacian eigenvalues <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \partial _1\ge \partial _2\ge \dots \partial _{n-1}>\partial _n=0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>∂</mi> <mn>1</mn> </msub> <mo>≥</mo> <msub> <mi>∂</mi> <mn>2</mn> </msub> <mo>≥</mo> <mo>⋯</mo> <msub> <mi>∂</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>∂</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. For any real number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S_\beta (G)=\sum _{i=1}^{n-1}\partial ^\beta _i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>β</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>∂</mi> <mi>i</mi> <mi>β</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> be the sum of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-th powers of the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(n-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> largest distance Laplacian eigenvalues of G. In this paper, we obtain various bounds for the graph invariant <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S_\beta (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>β</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which relates it with different parameters associated to the structure of the graph. We also obtain bounds for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$S_\beta (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>β</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of the number of components of the complement graph <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\overline{G}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>G</mi> <mo>¯</mo> </mover> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

On the sum of powers of distance Laplacian eigenvalues in terms of Wiener index and complement of a graph

期刊论文
发表日期：

2025年2月18日

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Ummer Mushtaq,
S. Pirzada,
Mohammad Abrar Ul Haq,
Saleem Khan

摘要

Let G be a connected graph with n vertices, m edges and having distance Laplacian eigenvalues $ \partial _1\ge \partial _2\ge \dots \partial _{n-1}>\partial _n=0 $ $\partial_{1} \geq \partial_{2} \geq \dots \partial_{n - 1} > \partial_{n} = 0$ . For any real number $\beta \ne 0$ $β \neq 0$ , let $S_\beta (G)=\sum _{i=1}^{n-1}\partial ^\beta _i$ $S_{β} (G) = \sum_{i = 1}^{n - 1} \partial_{i}^{β}$ be the sum of $\beta $ $β$ -th powers of the $(n-1)$ $(n - 1)$ largest distance Laplacian eigenvalues of G. In this paper, we obtain various bounds for the graph invariant $S_\beta (G)$ $S_{β} (G)$ , which relates it with different parameters associated to the structure of the graph. We also obtain bounds for $S_\beta (G)$ $S_{β} (G)$ in terms of the number of components of the complement graph $\overline{G}.$ $\bar{G} .$