On Mostar and Edge Mostar Indices of Graphs
Let G be a graph with edge set EG and e=uv∈EG. Define nue,G and mue,G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v, respectively. The numbers nve,G and mve,G can be defined in an analogous way. The Mostar and edge Mostar indices of G are...
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2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/6651220 |
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| author | Ali Ghalavand Ali Reza Ashrafi Mardjan Hakimi-Nezhaad |
| author_facet | Ali Ghalavand Ali Reza Ashrafi Mardjan Hakimi-Nezhaad |
| author_sort | Ali Ghalavand |
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| description | Let G be a graph with edge set EG and e=uv∈EG. Define nue,G and mue,G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v, respectively. The numbers nve,G and mve,G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as MoG=∑uv∈EGnuuv,G−nvuv,G and MoeG=∑uv∈EGmuuv,G−mvuv,G, respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented. |
| format | Article |
| id | doaj-art-acb9285c1d51414eb6769048f6bf03b6 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-acb9285c1d51414eb6769048f6bf03b62025-02-03T01:05:27ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/66512206651220On Mostar and Edge Mostar Indices of GraphsAli Ghalavand0Ali Reza Ashrafi1Mardjan Hakimi-Nezhaad2Department of Pure Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan 87317-53153, IranDepartment of Pure Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan 87317-53153, IranDepartment of Pure Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan 87317-53153, IranLet G be a graph with edge set EG and e=uv∈EG. Define nue,G and mue,G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v, respectively. The numbers nve,G and mve,G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as MoG=∑uv∈EGnuuv,G−nvuv,G and MoeG=∑uv∈EGmuuv,G−mvuv,G, respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented.http://dx.doi.org/10.1155/2021/6651220 |
| spellingShingle | Ali Ghalavand Ali Reza Ashrafi Mardjan Hakimi-Nezhaad On Mostar and Edge Mostar Indices of Graphs Journal of Mathematics |
| title | On Mostar and Edge Mostar Indices of Graphs |
| title_full | On Mostar and Edge Mostar Indices of Graphs |
| title_fullStr | On Mostar and Edge Mostar Indices of Graphs |
| title_full_unstemmed | On Mostar and Edge Mostar Indices of Graphs |
| title_short | On Mostar and Edge Mostar Indices of Graphs |
| title_sort | on mostar and edge mostar indices of graphs |
| url | http://dx.doi.org/10.1155/2021/6651220 |
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