On the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue of a graph
| Autor: | Hanyuan Deng, S. Balachandran, S. K. Ayyaswamy, Y. B. Venkatakrishnan |
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| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | Transactions on Combinatorics, Vol 6, Iss 4, Pp 43-50 (2017) |
| Druh dokumentu: | article |
| ISSN: | 2251-8657 2251-8665 |
| DOI: | 10.22108/toc.2017.21470 |
| Popis: | The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $eccleft(Gright)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The harmonic index $Hleft(Gright)$ of a graph $G$ is defined as the sum of $frac{2}{d_{i}+d_{j}}$ over all edges $v_{i}v_{j}$ of $G$, where $d_{i}$ denotes the degree of a vertex $v_{i}$ in $G$. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of $n$-vertex trees with two adjacent vertices of maximum degree $Delta$, where $ngeq 2Delta$. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randi'{c} index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas cite{hl}. |
| Databáze: | Directory of Open Access Journals |
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