On the Randić index of graphs
For a given graph G = (V, E), the degree mean rate of an edge uv ∈ E is a half of the quotient between the geometric and arithmetic means of its end-vertex degrees d(u) and d(v). In this note, we derive tight bounds for the Randić index of G in terms of its maximum and minimum degree mean rates over...
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| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2018 |
| País: | España |
| Recursos: | Universitat de Lleida (UdL) |
| Repositorio: | Repositori Obert UdL |
| OAI Identifier: | oai:repositori.udl.cat:10459.1/66759 |
| Acesso em linha: | https://doi.org/10.1016/j.disc.2018.08.020 http://hdl.handle.net/10459.1/66759 |
| Access Level: | acceso abierto |
| Palavra-chave: | Edge degree rate Randić index Connectivity index Mean distance |
| Resumo: | For a given graph G = (V, E), the degree mean rate of an edge uv ∈ E is a half of the quotient between the geometric and arithmetic means of its end-vertex degrees d(u) and d(v). In this note, we derive tight bounds for the Randić index of G in terms of its maximum and minimum degree mean rates over its edges. As a consequence, we prove the known conjecture that the average distance is bounded above by the Randić index for graphs with order n large enough, when the minimum degree δ is greater than (approximately) ∆1/3 , where ∆ is the maximum degree. As a by-product, this proves that almost all random (Erdös-Rényi) graphs satisfy the conjecture. |
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