Geodesic
The Degree-Diameter Problem for Outerplanar Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 823-834

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For positive integers Δ and D we define n_Δ, D to be the largest number of vertices in an outerplanar graph of given maximum degree Δ and diameter D. We prove that n_Δ , D = Δ^D/2 + O ( Δ^ D/2 -1 ) if D is even, and n_Δ, D = 3 Δ^D−1/2 + O (Δ^D−1/2−1 ) if D is odd. We then extend our result to maximal outerplanar graphs by showing that the maximum number of vertices in a maximal outerplanar graph of maximum degree Δ and diameter D asymptotically equalsn_Δ , D .
Keywords: outerplanar, diameter, degree, degree-diameter problem, distance, separator theorem 
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Dankelmann, Peter; Jonck, Elizabeth; Vetrík, Tomáš. The Degree-Diameter Problem for Outerplanar Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 823-834. http://geodesic.mathdoc.fr/item/DMGT_2017_37_3_a22/