Fixed Point Theorem on High Dimensional Apollonian Networks

This paper highlights a fixed point theorem on the subset constructed by growing d-dimensional Apollonian graph sequence, for any d from N\{0, 1}. We base our paper on a weighted graph edit distance constructing a metric space on the set of high dimensional Apollonian networks. The main result of this paper shows that we can select a subset of a growing graph sequence on the set of d-dimensional Apollonian graphs such that it's limit is an unique fixed point. The sequence is constructed using an IFS, which gives a growing Apollonian graph sequence. As examples, we highlight the fixed point limit of the two and three dimensional cases filling the inner of the regular triangle and the regular tetrahedron, respectively.