Sombor index of maximal outerplanar graphs (Q6585248)

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 \(\operatorname{SO}(G) = \sum_{xy \in E(G)} \sqrt{d^2(x) + d^2(y)}\).\N\NIn this paper, the authors prove that if \(G\) is a maximal outerplanar graph of order \(n\), then\N\[\N\operatorname{SO}(G) \geq 4(2n - 11)\sqrt{2}+ 2 \sqrt{13} + 4 \sqrt{5} + 20\N\]\Nand\N\[\N\operatorname{SO}(G) \leq 2 \sqrt{(n - 1)^2 + 4} + (n - 3) \sqrt{(n - 1)^2 + 9} + 3(n - 4) \sqrt{2} + 2 \sqrt{13}.\N\]\NThe authors also characterize the maximal outerplanar graphs achieving the above lower and upper bounds of the Somber index.