Convexity result and trees with large Balaban index
Balaban index is defined as J(G)=mm−n+2Σ1w(u)⋅w(v),$J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Sciendo
2018-08-01
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| Series: | Applied Mathematics and Nonlinear Sciences |
| Online Access: | https://doi.org/10.21042/AMNS.2018.2.00034 |
| _version_ | 1818744708565827584 |
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| author | Knor Martin Škrekovski Riste Tepeh Aleksandra |
| author_facet | Knor Martin Škrekovski Riste Tepeh Aleksandra |
| author_sort | Knor Martin |
| collection | DOAJ |
| description | Balaban index is defined as J(G)=mm−n+2Σ1w(u)⋅w(v),$J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w(u) (resp. w(v)) denotes the sum of distances from u (resp. v) to all the other vertices of G. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each n, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture. |
| first_indexed | 2024-12-18T02:48:36Z |
| format | Article |
| id | doaj.art-6533e4ca4632417b976b6217c71f3150 |
| institution | Directory Open Access Journal |
| issn | 2444-8656 |
| language | English |
| last_indexed | 2024-12-18T02:48:36Z |
| publishDate | 2018-08-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Applied Mathematics and Nonlinear Sciences |
| spelling | doaj.art-6533e4ca4632417b976b6217c71f31502022-12-21T21:23:31ZengSciendoApplied Mathematics and Nonlinear Sciences2444-86562018-08-013243344610.21042/AMNS.2018.2.00034Convexity result and trees with large Balaban indexKnor Martin0Škrekovski Riste1Tepeh Aleksandra2Slovak University of Technology in Bratislava, Faculty of Civil Engineering, Department of Mathematics, Radlinského 11, 813 68, Bratislava, SlovakiaFaculty of Information Studies, 8000Novo Mesto, SloveniaFaculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova ulica 17, 2000Maribor, SloveniaBalaban index is defined as J(G)=mm−n+2Σ1w(u)⋅w(v),$J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w(u) (resp. w(v)) denotes the sum of distances from u (resp. v) to all the other vertices of G. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each n, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.https://doi.org/10.21042/AMNS.2018.2.00034 |
| spellingShingle | Knor Martin Škrekovski Riste Tepeh Aleksandra Convexity result and trees with large Balaban index Applied Mathematics and Nonlinear Sciences |
| title | Convexity result and trees with large Balaban index |
| title_full | Convexity result and trees with large Balaban index |
| title_fullStr | Convexity result and trees with large Balaban index |
| title_full_unstemmed | Convexity result and trees with large Balaban index |
| title_short | Convexity result and trees with large Balaban index |
| title_sort | convexity result and trees with large balaban index |
| url | https://doi.org/10.21042/AMNS.2018.2.00034 |
| work_keys_str_mv | AT knormartin convexityresultandtreeswithlargebalabanindex AT skrekovskiriste convexityresultandtreeswithlargebalabanindex AT tepehaleksandra convexityresultandtreeswithlargebalabanindex |