Exact solutions for geodesic distance on treelike models with some constraints

Geodesic distance, commonly called shortest path length, has proved useful in a great variety of disciplines. It has been playing a significant role in search engine at present and so attracted considerable attention at the last few decades, particularly, almost all data structures and corresponding algorithms suitable to searching information generated based on treelike models. Hence, we, in this paper, study in detail geodesic distance on some treelike models which can be generated by three different types of operations, including first-order subdivision, ($1,m$)-star-fractal operation and $m$-vertex-operation. Compared to the most best used approaches for calculating geodesic distance on graphs, for instance, enumeration method and matrix multiplication, we take useful advantage of a novel method consisting in spirit of the concept of vertex cover in the language of graph theory and mapping. For each kind of treelike model addressed here, we certainly obtain an exact solution for its geodesic distance using our method. With the help of computer simulations, we confirm that the analytical results are in perfect agreement with simulations. In addition, we also report some intriguing structure properties on treelike models of two types among them. The one obeys exponential degree distribution seen in many complex networks, by contrast, the other possesses all but leaf vertices with identical degree and shows more homogeneous topological structure than the former. Besides that, the both have, in some sense, self-similar feature but instead the latter exhibits fractal property.