Conjugate Laplacian eigenvalues of co-neighbour graphs
Let \(G\) be a simple graph of order \(n\). A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in \(G\) are co-neighbour vertices if they share the same neighbours. Clearly, if \(S\) is a set of pairwise co-neighbour vertices of a graph \(G\), then \(S\) is...
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| Дата: | 2022 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Lugansk National Taras Shevchenko University
2022
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1754 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543029870985216 |
|---|---|
| author | Paul, S. |
| author_facet | Paul, S. |
| author_sort | Paul, S. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2022-10-14T16:01:17Z |
| description | Let \(G\) be a simple graph of order \(n\). A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in \(G\) are co-neighbour vertices if they share the same neighbours. Clearly, if \(S\) is a set of pairwise co-neighbour vertices of a graph \(G\), then \(S\) is an independent set of \(G\). Let \(c = a + b\sqrt{m}\) and \(\overline{c} = a-b\sqrt{m}\), where \(a\) and \(b\) are two nonzero integers and \(m\) is a positive integer such that \(m\) is not a perfect square. In [M. Lepovi\'c, On conjugate adjacency matrices of a graph, Discrete Mathematics, 307, 730--738, 2007], the author defined the matrix \(A^c(G) = [c_{ij} ]_n\) to be the conjugate adjacency matrix of \(G,\) if \(c_{ij}=c\) for any two adjacent vertices \(i\) and \(j\), \(c_{ij}= \overline{c}\) for any two nonadjacent vertices \(i\) and \(j\), and \(c_{ij}= 0\) if \(i=j\). In [S. Paul, Conjugate Laplacian matrices of a graph, Discrete Mathematics, Algorithms and Applications, 10, 1850082, 2018], the author defined the conjugate Laplacian matrix of graphs and described various properties of its eigenvalues and eigenspaces. In this article, we determine certain properties of the conjugate Laplacian eigenvalues and the eigenvectors of a graph with co-neighbour vertices. |
| first_indexed | 2025-12-02T15:39:11Z |
| format | Article |
| id | admjournalluguniveduua-article-1754 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:39:11Z |
| publishDate | 2022 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-17542022-10-14T16:01:17Z Conjugate Laplacian eigenvalues of co-neighbour graphs Paul, S. conjugate Laplacian matrix, co-neighbour vertices 05C50, 05C05, 15A18 Let \(G\) be a simple graph of order \(n\). A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in \(G\) are co-neighbour vertices if they share the same neighbours. Clearly, if \(S\) is a set of pairwise co-neighbour vertices of a graph \(G\), then \(S\) is an independent set of \(G\). Let \(c = a + b\sqrt{m}\) and \(\overline{c} = a-b\sqrt{m}\), where \(a\) and \(b\) are two nonzero integers and \(m\) is a positive integer such that \(m\) is not a perfect square. In [M. Lepovi\'c, On conjugate adjacency matrices of a graph, Discrete Mathematics, 307, 730--738, 2007], the author defined the matrix \(A^c(G) = [c_{ij} ]_n\) to be the conjugate adjacency matrix of \(G,\) if \(c_{ij}=c\) for any two adjacent vertices \(i\) and \(j\), \(c_{ij}= \overline{c}\) for any two nonadjacent vertices \(i\) and \(j\), and \(c_{ij}= 0\) if \(i=j\). In [S. Paul, Conjugate Laplacian matrices of a graph, Discrete Mathematics, Algorithms and Applications, 10, 1850082, 2018], the author defined the conjugate Laplacian matrix of graphs and described various properties of its eigenvalues and eigenspaces. In this article, we determine certain properties of the conjugate Laplacian eigenvalues and the eigenvectors of a graph with co-neighbour vertices. Lugansk National Taras Shevchenko University 2022-10-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1754 10.12958/adm1754 Algebra and Discrete Mathematics; Vol 33, No 2 (2022) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1754/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1754/816 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1754/863 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1754/1010 Copyright (c) 2022 Algebra and Discrete Mathematics |
| spellingShingle | conjugate Laplacian matrix co-neighbour vertices 05C50 05C05 15A18 Paul, S. Conjugate Laplacian eigenvalues of co-neighbour graphs |
| title | Conjugate Laplacian eigenvalues of co-neighbour graphs |
| title_full | Conjugate Laplacian eigenvalues of co-neighbour graphs |
| title_fullStr | Conjugate Laplacian eigenvalues of co-neighbour graphs |
| title_full_unstemmed | Conjugate Laplacian eigenvalues of co-neighbour graphs |
| title_short | Conjugate Laplacian eigenvalues of co-neighbour graphs |
| title_sort | conjugate laplacian eigenvalues of co-neighbour graphs |
| topic | conjugate Laplacian matrix co-neighbour vertices 05C50 05C05 15A18 |
| topic_facet | conjugate Laplacian matrix co-neighbour vertices 05C50 05C05 15A18 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1754 |
| work_keys_str_mv | AT pauls conjugatelaplacianeigenvaluesofconeighbourgraphs |