A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices
The Wiener polarity index of a graph G, usually denoted by WpG, is defined as the number of unordered pairs of those vertices of G that are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a tree T is a nontrivial path S whose end-vertices have degr...
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| Format: | Article |
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Wiley
2021-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2021/1052927 |
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| author | Sadia Noureen Akhlaq Ahmad Bhatti Akbar Ali |
| author_facet | Sadia Noureen Akhlaq Ahmad Bhatti Akbar Ali |
| author_sort | Sadia Noureen |
| collection | DOAJ |
| description | The Wiener polarity index of a graph G, usually denoted by WpG, is defined as the number of unordered pairs of those vertices of G that are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a tree T is a nontrivial path S whose end-vertices have degrees different from 2 in T and every other vertex (if exists) of S has degree 2 in T. In this note, the best possible sharp lower bounds on the Wiener polarity index Wp are derived for the trees of fixed order and with a given number of branching vertices or segments, and all the trees attaining this lower bound are characterized. |
| format | Article |
| id | doaj-art-d6e10d43dddd4483b48e70c11bcfbd9d |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-d6e10d43dddd4483b48e70c11bcfbd9d2025-02-03T01:20:09ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2021-01-01202110.1155/2021/10529271052927A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching VerticesSadia Noureen0Akhlaq Ahmad Bhatti1Akbar Ali2Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, B-Block, Faisal Town, Lahore, PakistanDepartment of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, B-Block, Faisal Town, Lahore, PakistanDepartment of Mathematics, Faculty of Science, University of Ha’il, Ha’il, Saudi ArabiaThe Wiener polarity index of a graph G, usually denoted by WpG, is defined as the number of unordered pairs of those vertices of G that are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a tree T is a nontrivial path S whose end-vertices have degrees different from 2 in T and every other vertex (if exists) of S has degree 2 in T. In this note, the best possible sharp lower bounds on the Wiener polarity index Wp are derived for the trees of fixed order and with a given number of branching vertices or segments, and all the trees attaining this lower bound are characterized.http://dx.doi.org/10.1155/2021/1052927 |
| spellingShingle | Sadia Noureen Akhlaq Ahmad Bhatti Akbar Ali A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices Discrete Dynamics in Nature and Society |
| title | A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices |
| title_full | A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices |
| title_fullStr | A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices |
| title_full_unstemmed | A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices |
| title_short | A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices |
| title_sort | note on the minimum wiener polarity index of trees with a given number of vertices and segments or branching vertices |
| url | http://dx.doi.org/10.1155/2021/1052927 |
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