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:
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