Sombor index of maximal outerplanar graphs. (English) Zbl 07894669
Let \(G = (V(G), E(G))\) be a graph. The degree of a vertex \(v\) in \(G\) is denoted by \(d(v)\). The Somber index of the graph \(G\) is defined as \(SO(G) = \sum_{xy \in E(G)} \sqrt{d^2(x) + d^2(y)}\).
In this paper, the authors prove that if \(G\) is a maximal outerplanar graph of order \(n\), then \[ SO(G) \geq 4(2n - 11)\sqrt{2}+ 2 \sqrt{13} + 4 \sqrt{5} + 20 \] and \[ SO(G) \leq 2 \sqrt{(n - 1)^2 + 4} + (n - 3) \sqrt{(n - 1)^2 + 9} + 3(n - 4) \sqrt{2} + 2 \sqrt{13}. \] The authors also characterize the maximal outerplanar graphs achieving the above lower and upper bounds of the Somber index.
In this paper, the authors prove that if \(G\) is a maximal outerplanar graph of order \(n\), then \[ SO(G) \geq 4(2n - 11)\sqrt{2}+ 2 \sqrt{13} + 4 \sqrt{5} + 20 \] and \[ SO(G) \leq 2 \sqrt{(n - 1)^2 + 4} + (n - 3) \sqrt{(n - 1)^2 + 9} + 3(n - 4) \sqrt{2} + 2 \sqrt{13}. \] The authors also characterize the maximal outerplanar graphs achieving the above lower and upper bounds of the Somber index.
Reviewer: Rao Li (Aiken)
MSC:
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C92 | Chemical graph theory |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
| 05C35 | Extremal problems in graph theory |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
References:
| [1] | Bondy, J. A.; Murty, U. S.R., Graph Theory, 1-676, 2008, Springer: Springer Belin · Zbl 1134.05001 |
| [2] | Chen, H.; Li, W.; Wang, J., Extremal values on the sombor index of trees, MATCH Commun. Math. Comput. Chem., 87, 1, 23-49, 2022 · Zbl 1503.92084 |
| [3] | Cruz, R.; Rada, J., Extremal values of the sombor index in unicyclic and bicyclic graphs, J. Math. Chem., 59, 1098-1116, 2021 · Zbl 1462.05071 |
| [4] | Gutman, I., Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86, 1, 11-16, 2021 · Zbl 1474.92154 |
| [5] | Hou, A.; Li, S.; Song, L., Sharp bounds for zagreb indices of maximal outerplanar graphs, J. Comb. Optim., 22, 252-269, 2011 · Zbl 1250.90102 |
| [6] | Liu, H.; Chen, H.; Xiao, Q.; Fang, X.; Tang, Z., More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons, Int. J. Quantum Chem., 121, 17, 2021, \( \sharp\) e26689 |
| [7] | Liu, H.; Gutman, I.; You, L.; Huang, Y., Sombor index: review of extremal results and bounds, J. Math. Chem., 60, 771-798, 2022 · Zbl 1492.92156 |
| [8] | Redžepović, I., Chemical applicability of sombor indices, J. Serb. Chem. Soc., 86, 445-457, 2021 |
| [9] | Song, L.; Liu, H.; Tang, Z., On the total eccentricity index of maximal outerplanar graphs, J. Shaoyang College: Natural Sci. Ed., 17, 3, 6-16, 2020 |
| [10] | Su, G.; Meng, M.; Cui, L., The general zeroth-order randic index of maximal outerplanar graphs and trees with k maximum degree vertices, J. Sci. Asia, 43, 387-393, 2017 |
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