Number Of Subtrees Research Articles

Enumeration Of Subtrees Of Two Families Of Self-Similar Networks Based On Novel Two-Forest Dual Transformations

Abstract As a structural topological index, the number of subtrees has great significance for the analysis and design of hybrid locally reliable networks. In this paper, with generating function and introducing a novel two-forest dual transformation technique, we solve the subtree enumerating problems of two representatives of the self-similar networks, such as the hierarchical lattice and $(u,v)$-flower networks. Moreover, by means of the circle weight transfer technique, two linear time algorithms of computing the subtree generation functions of these two families of networks are also proposed. The subtree density of two special cases for these self-similar networks is briefly discussed as an application.

Computing the expected subtree number of random hexagonal and phenylene chains based on probability matrices

With structural analysis and probability matrix, this paper studies the subtree number index (number of non-labeled subtrees) of random hexagonal and phenylene chains, and presents exact recursive formulas for the expected values of subtree number index of the random hexagonal and phenylene chains in terms of probability matrices, as applications, we obtain the subtree number of hexagonal linear chain, hexagonal helicence chain, phenylene linear chain and phenylene helicence chain. Moreover, the proposed matrix method also builds a bridge to explore the deep correlation between structural-based and distance-based topological indices, and provides important theoretical support for prediction new properties of chemical compounds.

Broadcasting‐induced colorings of preferential attachment trees

AbstractWe consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.

Enumerating Subtrees of Flower and Sunflower Networks

Symmetry widely exists in many complex and real-world networks, with flower networks and sunflower networks being two richly symmetric networks and having many practical applications due to their special structures. The number of subtrees (the subtree number index) is closely related to the reliable network design. Using a generating function, structural analysis techniques, and auxiliary structure introduction, this paper presents the subtree generating functions of flower networks Fln,m(n≥3,m≥2) and sunflower networks Sfn,m(n≥3,m≥2) and, thus, solves the computation of subtree number indices of Fln,m(n≥3,m≥2) and Sfn,m(n≥3,m≥2). The results provide a fundamental and efficient method for exploring novel features of symmetric complex cyclic networks from the structural subtree number index perspective. For instance, we conclude that under some parameter constraints, the flower networks are more reliable than sunflower networks.

Structural Damage Identification System Suitable for Old Arch Bridge in Rural Regions: Random Forest Approach

A huge number of old arch bridges located in rural regions are at the peak of maintenance. The health monitoring technology of the long-span bridge is hardly applicable to the small-span bridge, owing to the absence of technical resources and sufficient funds in rural regions. There is an urgent need for an economical, fast, and accurate damage identification solution. The authors proposed a damage identification system of an old arch bridge implemented with a machine learning algorithm, which took the vehicle-induced response as the excitation. A damage index was defined based on wavelet packet theory, and a machine learning sample database collecting the denoised response was constructed. Through comparing three machine learning algorithms: Back-Propagation Neural Network (BPNN), Support Vector Machine (SVM), and Random Forest (R.F.), the R.F. damage identification model were found to have a better recognition ability. Finally, the Particle Swarm Optimization (PSO) algorithm was used to optimize the number of subtrees and split features of the R.F. model. The PSO optimized R.F. model was capable of the identification of different damage levels of old arch bridges with sensitive damage index. The proposed framework is practical and promising for the old bridge’s structural damage identification in rural regions.

A Conjecture on Different Central Parts of Binary Trees

Let \(\Omega _n\) be the family of binary trees on n vertices obtained by identifying the root of an rgood binary tree with a vertex of maximum eccentricity of a binary caterpillar. In the paper titled “On different middle parts of a tree" (The Electronic Journal of Combinatorics, 25 (2018), no. 3, paper 3.17, 32 pp), Smith et al. conjectured that among all binary trees on n vertices the pairwise distance between any two of center, centroid and subtree core is maximized by some member of the family \(\Omega _n\). We first obtain the rooted binary tree which minimizes the number of root-containing subtrees and then prove this conjecture. We also obtain the binary trees which maximize these distances.

Enumeration of subtrees of planar two-tree networks

The number of subtrees, also referred to as the subtrees index, is a key parameter to measure graph structures such as networks. In this paper, we investigate the number of subtrees of planar two-tree networks. By “adding a virtual edge” and “edge orientation”, we present a linear time algorithm for computing the number of subtrees of planar two-tree networks, as well as a family of planar two-connected networks. As applications, we provide the formulae for the number of subtrees of the famous small-world Farey network and GDURT network. We also discuss the relationship between the spanning subtree number and the subtree number of these networks.

On computing the number of (BC-)subtrees, eccentric subtree number, and global and local means of trees

In this paper, we present algorithms for computing the number of subtrees of trees in linear-time, the number of BC-subtrees of trees in linear-time, the global mean of trees in linear-time, the local mean of a given vertex of trees in linear-time, and the eccentric subtree number on a given vertex of trees in quadratic-time, where a BC-tree is a tree such that the distance between any two leaves is even, the eccentric subtree number on a vertex v in a given tree T is defined as the minimum number of subtrees of T such that each subtree contains both u and v for all u in T, and the global mean of trees is the average number of vertices in a subtree (averaged over all subtrees) and the local mean of a given vertex v in trees is the average number of vertices in a subtree that contains v (averaged over all subtrees that contain v).

Comparative results between the number of subtrees and Wiener index of graphs

For a graph G, we denote by N(G) the number of non-empty subtrees of G. If G is connected, its Wiener index W(G) is the sum of distances between all unordered pairs of vertices of G. In this paper we establish some comparative results between N and W. It is shown that N(G) > W(G) if G is a graph of order n ≥ 7 and diameter 2 or 3. Also some graphs are constructed with large diameters and N > W. Moreover, for a tree T ≇ Sn of order n, we prove that W(T) > N(T) if T is a starlike tree with maximum degree 3 or a tree with exactly two vertices of maximum degrees 3 one of which has two leaf neighbors, or a broom with klog2 n leaves. And a method is provided for constructing the graphs with N < W. Finally several related open problems are proposed to the comparison between N and W.

Maximum number of subtrees in cacti and block graphs

Maximum number of subtrees in cacti and block graphs

Research on visualisation transmission method for business innovation strategy data based on structural characteristics

In order to overcome the problems of low transfer accuracy and user satisfaction existing in the existing data visualisation transfer methods, this paper proposes a data visualisation transfer method of business innovation strategy based on structural features. The similarity calculation method of structure feature tree is used to realise the data structure feature processing. The sub-trees of all non-isomorphic patterns of K nodes are constructed. The number of isomorphic sub-trees is calculated in the tree, and the transfer data division is completed. Based on the basic program of business intelligence, the original data of business innovation strategy is processed, converted into various types of visualisation structure data, and transmitted to relevant personnel in online form to complete the visualisation transmission of business innovation strategy data. The experimental results show that the proposed method is more accurate, up to 99.3%, and users are more satisfied.

Generalised outerplanar Turán numbers and maximum number of [formula omitted]-vertex subtrees

We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle the generalised outerplanar Turán number for all cycles.We also determine the exponential growth of the generalised outerplanar Turán number of paths Pk as a function of k which implies the order of magnitude of the generalised outerplanar Turán number of arbitrary trees. The bounds are strongly related to the sequence of Catalan numbers.

The number of subtrees in graphs with given number of cut edges

The number of non-empty subtrees of a connected graph G, denoted by N(G), has been an interesting topic of research for many years. Considered as a counting based topological index, it appears to have correlations with many other well studied graph indices. Many results on various extremal trees with respect to the number of subtrees have been established over the years. But relatively few are available on more general graphs. In this paper we consider the maximum and minimum values of N(G) among connected graphs of a given order n and k≥0 cut edges. Sharp upper bounds are presented with the corresponding extremal structures for each k≥0, and sharp lower bounds are presented with the corresponding extremal structures for 0≤k<n−12.

On the probability that a random subtree is spanning

AbstractWe consider the quantity associated with a graph that is defined as the probability that a randomly chosen subtree of is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that is bounded below by a positive constant provided that the minimum degree is bounded below by a linear function in the number of vertices. Thereafter, the focus is shifted to the classical Erdős–Rényi random graph model . It is shown that converges in probability to if and to 0 if . As side results, we find in the dense case that the total number of subtrees satisfies a log‐normal limit law, and that the number of vertices missing in a random subtree asymptotically follows a Poisson distribution.

On sampling error in genetic programming

The initial population in genetic programming (GP) should form a representative sample of all possible solutions (the search space). While large populations accurately approximate the distribution of possible solutions, small populations tend to incorporate a sampling error. This paper analyzes how the size of a GP population affects the sampling error and contributes to answering the question of how to size initial GP populations. First, we present a probabilistic model of the expected number of subtrees for GP populations initialized with full, grow, or ramped half-and-half. Second, based on our frequency model, we present a model that estimates the sampling error for a given GP population size. We validate our models empirically and show that, compared to smaller population sizes, our recommended population sizes largely reduce the sampling error of measured fitness values. Increasing the population sizes even more, however, does not considerably reduce the sampling error of fitness values. Last, we recommend population sizes for some widely used benchmark problem instances that result in a low sampling error. A low sampling error at initialization is necessary (but not sufficient) for a reliable search since lowering the sampling error means that the overall random variations in a random sample are reduced. Our results indicate that sampling error is a severe problem for GP, making large initial population sizes necessary to obtain a low sampling error. Our model allows practitioners of GP to determine a minimum initial population size so that the sampling error is lower than a threshold, given a confidence level.

Extremal Trees with Fixed Degree Sequence

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

To Ameliorate Classification Accuracy using Ensemble Distributed Decision Tree (DDT) Vote Approach: An Empirical discourse of Geographical Data Mining

Abstract Weather data of Kashmir province has 6 attributes recorded at three different substations. This paper proposes a distributed decision tree algorithm and its implementation on Historical Geographical data of Kashmir province. The machine learning Decision tree algorithm applied on the Kashmir province dataset generates the accuracy of 81.54%. The distributed decision tree generates multiple trees based on the partitions of the original dataset in which the data is segregated according to the substations (42026, 42027 and 42044). The ratio between generated data sets was distributed in 32.38%, 34.19% and 33.42% respectively which is appropriate for the parallelism. Its distributed implementation, i.e. Distributed Decision Tree produces a specified number of sub-trees (depending upon number of partitions of input dataset) and at the end collects votes or averages the prediction or classification. In this paper, we have implemented the hard- voting approach to calculate the overall performance of the n-number of trees in distributed environment. The empirical results demonstrate that distributed decision trees approach has not improved the overall accuracy as compared to the original dataset without partitioning.

Entropy and Enumeration of Subtrees in a Cactus Network

For a given network, the number of spanning trees is a key parameter to measure its reliability in edge failure cases, while the number of subtrees is a key parameter to measure its reliability in both vertex and edge failures cases. Zhang et al. investigated the entropy of spanning trees, spanning forests and connected spanning subgraphs of Koch networks. In this paper, we extend Koch networks to 3-cactus networks, and study the entropy of subtrees of 3-cactus networks. We present a linear algorithm to count the number of subtrees in a 3-cactus network, determine the upper and lower bounds of the entropy of subtrees of these networks and characterize those attaining the extremal values. As an application, a linear algorithm is developed to count the number of subtrees in Koch networks, with complexity O(g), where g is the number of iterations. Finally, we determine the entropy of subtrees of Koch networks.

Dynamic reverse supply chain network design under uncertainty: mathematical modeling and solution algorithm

AbstractMotivated by the recovery of modular‐structured products, this study addresses the flexible design of a reverse supply chain (RSC) over a planning horizon while incorporating the dynamic uncertain behavior of product returns. The stochastic parameter is modeled as a scenario tree and therefore the concerned problem is formulated as a multistage mixed‐integer stochastic program. To alleviate the computational complexity of the proposed model, it is decomposed into smaller scenario cluster submodels associated with a number of subtrees that share a certain number of predecessor nodes in the original scenario tree. The submodels are coordinated into an implementable solution via a Lagrangian‐progressive hedging‐based method that employs a viable Benders decomposition based algorithm for solving each scenario cluster submodel. Based on a realistic scale case, computational results indicate the superiority of the proposed flexible dynamic RSC design model compared to the existing models. Results also demonstrate the efficiency of the proposed solution approach.

Algorithms Based on Path Contraction Carrying Weights for Enumerating Subtrees of Tricyclic Graphs

Abstract The subtree number index of a graph, defined as the number of subtrees, attracts much attention recently. Finding a proper algorithm to compute this index is an important but difficult problem for a general graph. Even for unicyclic and bicyclic graphs, it is not completely trivial, though it can be figured out by try and error. However, it is complicated for tricyclic graphs. This paper proposes path contraction carrying weights (PCCWs) algorithms to compute the subtree number index for the nontrivial case of bicyclic graphs and all 15 cases of tricyclic graphs, based on three techniques: PCCWs, generating function and structural decomposition. Our approach provides a foundation and useful methods to compute subtree number index for graphs with more complicated cycle structures and can be applied to investigate the novel structural property of some important nanomaterials such as the pentagonal carbon nanocone.