On Extremal Values of the Nk-Degree Distance Index in Trees

The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index (k-distance degr...

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Main Authors:	Zia Ullah Khan, Quaid Iqbal
Format:	Article
Language:	English
Published:	MDPI AG 2025-07-01
Series:	Mathematics
Subjects:	distance topological index <i>N<sub>k</sub></i>-index trees broom graph distances in graphs
Online Access:	https://www.mdpi.com/2227-7390/13/14/2284
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author	Zia Ullah Khan Quaid Iqbal
author_facet	Zia Ullah Khan Quaid Iqbal
author_sort	Zia Ullah Khan
collection	DOAJ
description	The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index (<i>k</i>-distance degree index) of a connected graph <i>G</i> was first introduced by Naji and Soner as a generalization of the distance degree concept, as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munderover></mstyle><mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder></mstyle><msub><mi>d</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mfenced><mi>k</mi></mrow></semantics></math></inline-formula>, where the distance between <i>u</i> and <i>v</i> in <i>G</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the diameter of a graph <i>G</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and the degree of a vertex <i>v</i> at distance <i>k</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mi>d</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mspace width="4pt"></mspace><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we extend the study of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index of graphs. We introduced some graph transformations and their impact on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index of graph and proved that the star graph has the minimum, and the path graph has the maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index among the set of all trees on <i>n</i> vertices. We also show that among all trees with fixed maximum-degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>, the broom graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mrow><mi>n</mi><mo>,</mo><mo>Δ</mo></mrow></msub></semantics></math></inline-formula> (consisting of a star <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mrow><mo>Δ</mo><mo>+</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> and a pendant path of length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mo>Δ</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> attached to any arbitrary pendant path of star) is a unique tree which maximizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index. Further, we also defined and proved a graph with maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index for a given number of <i>n</i> vertices, maximum-degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>, and perfect matching among trees. We characterize the starlike trees which minimize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index and propose a unique tree which minimizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index with diameter <i>d</i> and <i>n</i> vertices among trees.
format	Article
id	doaj-art-c4b3a187074e4c4da8ceae253f8245e5
institution	Kabale University
issn	2227-7390
language	English
publishDate	2025-07-01
publisher	MDPI AG
record_format	Article
series	Mathematics
spelling	doaj-art-c4b3a187074e4c4da8ceae253f8245e52025-08-20T03:58:30ZengMDPI AGMathematics2227-73902025-07-011314228410.3390/math13142284On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in TreesZia Ullah Khan0Quaid Iqbal1School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, ChinaCenter for Combinatorics, Nankai University, Tianjin 300071, ChinaThe <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index (<i>k</i>-distance degree index) of a connected graph <i>G</i> was first introduced by Naji and Soner as a generalization of the distance degree concept, as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munderover></mstyle><mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder></mstyle><msub><mi>d</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mfenced><mi>k</mi></mrow></semantics></math></inline-formula>, where the distance between <i>u</i> and <i>v</i> in <i>G</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the diameter of a graph <i>G</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and the degree of a vertex <i>v</i> at distance <i>k</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mi>d</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mspace width="4pt"></mspace><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we extend the study of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index of graphs. We introduced some graph transformations and their impact on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index of graph and proved that the star graph has the minimum, and the path graph has the maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index among the set of all trees on <i>n</i> vertices. We also show that among all trees with fixed maximum-degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>, the broom graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mrow><mi>n</mi><mo>,</mo><mo>Δ</mo></mrow></msub></semantics></math></inline-formula> (consisting of a star <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mrow><mo>Δ</mo><mo>+</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> and a pendant path of length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mo>Δ</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> attached to any arbitrary pendant path of star) is a unique tree which maximizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index. Further, we also defined and proved a graph with maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index for a given number of <i>n</i> vertices, maximum-degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>, and perfect matching among trees. We characterize the starlike trees which minimize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index and propose a unique tree which minimizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index with diameter <i>d</i> and <i>n</i> vertices among trees.https://www.mdpi.com/2227-7390/13/14/2284distance topological index<i>N<sub>k</sub></i>-indextreesbroom graphdistances in graphs
spellingShingle	Zia Ullah Khan Quaid Iqbal On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees Mathematics distance topological index <i>N<sub>k</sub></i>-index trees broom graph distances in graphs
title	On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees
title_full	On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees
title_fullStr	On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees
title_full_unstemmed	On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees
title_short	On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees
title_sort	on extremal values of the i n sub k sub i degree distance index in trees
topic	distance topological index <i>N<sub>k</sub></i>-index trees broom graph distances in graphs
url	https://www.mdpi.com/2227-7390/13/14/2284
work_keys_str_mv	AT ziaullahkhan onextremalvaluesoftheinsubksubidegreedistanceindexintrees AT quaidiqbal onextremalvaluesoftheinsubksubidegreedistanceindexintrees

On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees

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