Arxiv 

"High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks", 2nd Version
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* Added several references including Albenque & Marckert which proves the logarithmic lower bound of the diameter (references and diameter changes).
* Coupling and last section changed (simplifications).
* Corrected a bunch of typos (typos).
* Moved calculations of Lemma 2 in the Appendix.