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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.

Tree, as the simplest and most fundamental connected graph, has received consideration attention from a variety of disciplines. In this paper, our aim is to discuss a family of trees of great interest which are in fact divided into two groups. Unlike preexisting research focusing mainly on a single edge as seed, our treelike models are constructed by arbitrary tree T . Therefore, the first group contains trees generated based on tree T using first-order subdivision. The other is constituted by trees created from tree T with (1, m)-star-fractal operation. By the novel methods addressed shortly, we do capture analytically the exact solution for geodesic distance on each member in tree family of this type. Compared to some commonly adopted methods, for instance, Laplacian spectral, our techniques are much lighter to implement according to both generality and complexity. In addition, the closed-form expression of mean first-passage time (M F P T ) for random walk on each member is also readily obtained on the basis of our methods. Our results suggest that the two topological operations are sharply different from each other, particularly, M F P T for random walks, and however have likely to show the same function, at least, on average geodesic distance.

There is little attention paid to stochastic tree networks in comparison with the corresponding deterministic analogs in the current study of fractal trees. In this paper, we propose a principled framework for producing a family of stochastic growth tree networks [Formula: see text] possessing fractal characteristic, where [Formula: see text] represents the time step and parameter [Formula: see text] is the number of vertices newly created for each existing vertex at generation. To this end, we introduce two types of generative ways, i.e., Edge-Operation and Edge-Vertex-Operation. More interestingly, the resulting stochastic trees turn out to have an identical fractal dimension [Formula: see text] regardless of the introduction of randomness in the growth process. At the same time, we also study many other structural parameters including diameter and degree distribution. In both extreme cases, our tree networks are deterministic and follow multiple-point degree distribution and power-law degree distribution, respectively. Additionally, we consider random walks on stochastic growth tree networks [Formula: see text] and derive an expectation estimation for mean hitting time [Formula: see text] in an effective combinatorial manner instead of commonly used spectral methods. The result shows that on average, the scaling of mean hitting time [Formula: see text] obeys [Formula: see text], where [Formula: see text] represents vertex number and exponent [Formula: see text] is equivalent to [Formula: see text]. In the meantime, we conduct extensive experimental simulations and observe that empirical analysis is in strong agreement with theoretical results.