Narrow Results

Using the discrete version of finite pattern, we obtain the Wiener sum of fractal network modeled by the level-3 Sierpinski gasket.

The Fermat point of a triangle is the point with the minimal total distance from the three vertices of the triangle. Meanwhile, the total distance from the three vertices to the Fermat point is called the Fermat distance. In this paper, we discuss the Fermat distance on some networks. We study the mean Fermat distance of an unweighted and undirected hierarchical network $G_n$, which is obtained by analytical method and iterative calculation. We then reveal the relation between the mean Fermat distance $\overline{{ℱ}}$ and the mean geodesic distance $\bar{d}$ in general networks, namely, $\frac{3}{2}\bar{d}\le \overline{{ℱ}}\le 2\bar{d}$. The ratio of the mean Fermat distance to the mean geodesic distance of $G_n$ tends to $3/2$, which is the lower bound of the inequality above. Moreover, the result shows the small-world effect of $G_n$. Finally, we illustrate that the mean Fermat distance is significant in both small-world and scale-free networks.

In this paper, we investigate the average geodesic distance on the Sierpinski hexagon in terms of finite patterns on integrals. Applying this result, we also obtain the asymptotic formula for average distances of Sierpinski hexagon networks.

In this paper, we discuss a family of non-p.c.f. self-similar networks. Although the boundary of each fractal piece is not a finite set, we obtain the finite geometric patterns for the integral of geodesic distance on the self-similar measure, and then calculate its average distance.

Let $\mathcal{ℱ}$ be a set of countable many functions, where every function $f\in \mathcal{ℱ}$ satisfies $f:\mathbb{Z}\to \mathbb{Z}$ and $f(n)\to +\infty$ as $n\to +\infty$. This paper studies the size of the set consisting of those numbers whose Lüroth digit sequence is strictly increasing and contains any finite pattern of $\mathcal{ℱ}$. We prove that the Hausdorff dimension of such set is $1/2$ and give several applications.

In this paper, we discuss a family of p.c.f. self-similar fractal networks which have reflection transformations. We obtain the average geodesic distance on the corresponding fractal in terms of finite pattern of integrals. With these results, we also obtain the asymptotic formula for average distances of the skeleton networks.

The stretched Cantor product is not self-similar but rectifiable. Using the method of finite pattern, we study the mean geodesic distance on the stretched Cantor product.

In this paper, we establish the theory of finite geodesic patterns on limit space to calculate the average distances of touching networks.

The Fermat problem of triangle is interesting in plane geometry. Considering the Fermat problem within a fractal, we study the average Fermat distance of three random points on a self-similar tree by using the method of finite pattern.

The Fermat point of a triangle is a point that the total distance from the three vertices of the triangle to the point is the minimum possible. Considering the Fermat problem within a network, we study the average Fermat distance of three random nodes on the Vicsek networks by using the method of finite pattern.

The four-point Steiner distance is the minimum of the total geodesic distances within the metric space from four given points to a point. For Vicsek networks, we obtain the asymptotic formula of their mean Steiner distances using the method of finite pattern.

In this paper, we investigate a class of self-similar networks modeled on a self-similar fractal tree, and use the self-similar measure and the method of finite pattern to obtain the asymptotic formula of node-weighted average Fermat distances on fractal tree networks.

The point within a triangle that meets the minimum sum of the distances to the three vertices is called the Fermat point. The aim of this paper is to obtain the Fermat average distance on a self-similar fractal tree in terms of the self-similar measure and the technique of finite pattern.

The hyper-Wiener index is an important topological invariant, which is based on the distance between two vertexes on a graph. This paper is concerned with the hyper-Wiener indices of Vicsek networks. Based on the method of finite pattern, we calculate the asymptotic formula of the hyper-Wiener indices of Vicsek networks.

In this paper, we introduce the fractal version of Zagreb eccentricity index which is a singular integral of the eccentricity squared in terms of self-similar measure. Based on the geometric characteristics of the farthest point, we calculate the fractal version of Zagreb eccentricity index on the (3,1) flower fractal to be $\frac{1829}{360}$ by using the technique of finite pattern.

In this paper, we study Vicsek polygons and extended Vicsek networks, which are an extension of Vicsek fractal. Our research indices are some Fermat-type indices, including the Fermat eccentricity and the Fermat eccentric distance sum. Fermat-type indices are novel graph invariants with vast potential in research on structure–activity and quantitative structure–property. By the approach of finite pattern, we solve some integrals to gain their asymptotic formulas on Fermat eccentricity and Fermat eccentric distance sum.

The aim of this paper is to illustrate a strategy for how to get the exact formula of the distance sums of fractal networks (for example, the Sierpiń ski skeleton networks) with the technique of finite pattern [S. Wang, Z. Yu and L. Xi, Average geodesic distance of Sierpinski gasket and Sierpinski networks, Fractals 25(5) (2017) 1750044].

In this paper, we investigate the Zagreb eccentricity index of the level-$n$ Sierpinski gasket $K_n$. Based on the self-similarity and finite pattern, we compute the Zagreb eccentricity index of $K_n,$ which is the integral of square of eccentricity in terms of the self-similar singular measure.

The hyper-Wiener index plays an important role in chemical graph theory. In this paper, using the technique named finite pattern, we discuss the hyper-Wiener index on level-3 Sierpinski gasket which is a self-similar fractal.

The edge-Wiener index, an invariant index representing the summation of the distances between every pair of edges in the graph, has monumental influence on the study of chemistry and materials science. In this paper, drawing inspiration from Gromov’s idea, we use the finite pattern method proposed by Wang et al. [Average geodesic distance of Sierpinski gasket and Sierpinski networks, Fractals 25(5) (2017) 1750044] to figure out the exact formula of edge-Wiener index of the Sierpinski fractal networks.