Minimizing Degree-based Topological Indices for Trees with Given Number of Pendent Vertices + Erratum
| Autor: | Goubko, Mikhail, Réti, Tamás |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | MATCH Commun. Math. Comput. Chem. 71 (1) (2014) 33-46 + Erratum in V. 72 (3) (2014) 633-639 |
| Druh dokumentu: | Working Paper |
| Popis: | We derive sharp lower bounds for the first and the second Zagreb indices ($M_1$ and $M_2$ respectively) for trees and chemical trees with the given number of pendent vertices and find optimal trees. $M_1$ is minimized by a tree with all internal vertices having degree 4, while $M_2$ is minimized by a tree where each "stem" vertex is incident to 3 or 4 pendent vertices and one internal vertex, while the rest internal vertices are incident to 3 other internal vertices. The technique is shown to generalize to the weighted first Zagreb index, the zeroth order general Randi\'{c} index, as long as to many other degree-based indices. Later the erratum was added: Theorem 3 says that the second Zagreb index $M_2$ cannot be less than $11n-27$ for a tree with $n\ge 8$ pendent vertices. Yet the tree exists with $n=8$ vertices (the two-sided broom) violating this inequality. The reason is that the proof of Theorem 3 relays on a tacit assumption that an index-minimizing tree contains no vertices of degree 2. This assumption appears to be invalid in general. In this erratum we show that the inequality $M_2 \ge 11n-27$ still holds for trees with $n\ge 9$ vertices and provide the valid proof of the (corrected) Theorem 3. Comment: Original paper: 15 pages, 2 figures. Erratum: 8 pages, 2 figures. Professor Tam'as R'eti contributed to the erratum |
| Databáze: | arXiv |
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