Index Tree Research Articles

On Sombor index of trees

This paper is concerned with the recently introduced Sombor index SO, defined asSO=SO(G)=∑vkvℓ∈E(G)dG(vk)2+dG(vℓ)2,where dG(v) is the degree of the vertex v of a graph G. We present bounds on SO of trees in terms of order, independence number, and number of pendent vertices, and characterize the extremal cases. In addition, analogous results for quasi-trees are established.

Shallow population structure of southern bluefin tuna, Thunnus maccoyii (Castelnau, 1872) between Indian and Atlantic Oceans inferred from mitochondrial DNA sequences

Southern bluefin tuna (Thunnus maccoyii Castelnau, 1872) is distributed across most of the southern temperate ocean and migrates extensively between 30°S and 50°S. Since T. maccoyii has been continually and heavily exploited, it is necessary to investigate the genetic diversity, population structure and demographic history of T. maccoyii for effective management and conservation. Thirty-seven gonad tissues of T. maccoyii were sampled from two locations, which were in the eastern Indian Ocean and the eastern Atlantic Ocean, by scientific observers onboard Korean T. maccoyii longline vessels in 2015. We compared 1240-bp sequences of combined mitochondrial DNA (mtDNA) from the cytochrome c oxidase subunit I (COI, 504-bp) and control region (CR, 736-bp) sequences. The pairwise fixation index (F ST) and maximum-likelihood tree showed that two clades (A and B) were formed regardless of locations. Clade A occurred more commonly than clade B in both localities: the occurrence ratio of clade A was 69% in the Indian Ocean, and 79% in the Atlantic Ocean, respectively. Our findings suggest that a historic differentiation event may have occurred in T. maccoyii, but recently the connectivity between the two oceans may be possible in T. maccoyii populations.

Entire Wiener index of graphs

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph $G$ is equal to the sum of distances between all pairs of vertices of $G$. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this new index for subdivision and $k$-subdivision graphs and some graph operations are obtained.

Partial 3D object retrieval using local binary QUICCI descriptors and dissimilarity tree indexing

A complete pipeline is presented for accurate and efficient partial 3D object retrieval based on Quick Intersection Count Change Image (QUICCI) binary local descriptors and a novel indexing tree. It is shown how a modification to the QUICCI query descriptor makes it ideal for partial retrieval. An indexing structure called Dissimilarity Tree is proposed which can significantly accelerate searching the large space of local descriptors; this is applicable to QUICCI and other binary descriptors. The index exploits the distribution of bits within descriptors for efficient retrieval. The retrieval pipeline is tested on the artificial part of SHREC’16 dataset with near-ideal retrieval results.

On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

Estimating leaf area index of the Yellowwood tree (Podocarpus spp.) in an indigenous Southern African Forest, using Sentinel 2 Multispectral Instrument data and the Random Forest regression ensemble

Yellowwood or Podocarpus (spp.) holds the esteemed biodiversity status, as key forest species in the mist-belt Afromontane forests of southern Africa. The podocarps are listed as endangered species owing to extensive logging. The forest species support large communities of plants and birds, attributing to the maintenance of biodiversity. Therefore, there is a need to understand the condition of such keystone species if effective and comprehensive biodiversity conservation measures are to be drawn for these dwindling forests. Leaf area index is a crucial eco-physiological parameter applied in the evaluation of the growth and productivity of forest trees, hence it is a suitable proxy for understanding the condition of Yellowwood trees. This study, therefore, sought to estimate the leaf area index of the Yellowwood spp. using Sentinel 2 Multispectral instrument (S2 MSI) data in concert with the Random Forest regression ensemble. Specifically, individual wavebands and vegetation indices were used in developing leaf area index prediction models based on two approaches. The multistage approach, categorised the predictors according to the generalised order of progression, from standard spectral bands to vegetation indices. The second approach involved using a pooled set of predictors, with the backward elimination of poorly performing wavebands and vegetation indices. Results showed that the backward elimination method produced a better model (R2 = 0.59; RMSE = 0.48) when compared to the multistage approach (R2 = 0.50; RMSE = 0.48). The most influential predictor variables in both models were Band 5 and NDVI Red Edge 2. Results of this study underscore the prospects of Sentinel 2 MSI data in characterising the productivity of critical forest species such as the Yellowwoods of the Afromontane forest in southern Africa. The findings of this study are a fundamental step towards understanding forest health and productivity, required in deriving comprehensive monitoring and management strategies in biodiversity conservation.

Extreme Wiener indices of trees with given number of vertices of maximum degree

In this paper, we consider trees of order n with a given number of vertices of maximum degree and solve the minimal extremal problem for the Wiener index on that class. Characteristics of the extremal structures are presented, some directly from previously established methods. We also consider the maximal value of the Wiener index in the class of trees of order n with exactly k vertices of a given maximum degree, and characterize appropriate optimal trees.

Inverse Problem for the Forgotten and the Hyper Zagreb Indices of Trees

Let $G=(E(G),V(G))$ be a (molecular) graph with vertex set $V(G)$ and edge set $E(G)$. The forgotten Zagreb index and the hyper Zagreb index of G are defined by $F(G) = sum_{u in V(G)} d(u)^{3}$ and $HM(G) = sum_{uv in E(G)}(d(u)+d(v))^{2}$ where $d(u)$ and d(v) are the degrees of the vertices $u$ and $v$ in $G$, respectively. A recent problem called the inverse problem deals with the numerical realizations of topological indices. We see that there exist trees for all even positive integers with $F(G)>88$ and with $HM(G)>158$. Along with the result, we show that there exist no trees with $F(G) < 90$ and $HM(G) < 160$ with some exceptional even positive integers and hence characterize the forgotten Zagreb index and the hyper Zagreb index for trees.

Sharp lower bounds on the sum-connectivity index of trees

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. In this paper, some extremal problems on the sum-connectivity index of trees are studied. The extremal values on the sum-connectivity index of trees with given graphic parameters, such as pendant number, matching number, domination number and diameter, are determined. The corresponding extremal graphs are characterized, respectively.

Dynamic secure multi-keyword ranked search over encrypted cloud data

Due to increasing volume of stored data in organizations, data owners upload their data to cloud servers. Sensitive data has to be encrypted prior to uploading to the cloud server because of security requirements. Therefore, capable of searching over the encrypted cloud data is a very important issue. One of the most important challenges that must be addressed is to search whole documents to find related documents to a query. In this paper, a scheme is suggested to eliminate the need of searching for whole documents. A secure multi-keyword ranked search method over the encrypted cloud data is introduced that provides dynamic operation on the document collection. Also, access control mechanism is considered in the proposed scheme. The vector space model is employed to B+tree index and query creating. Due to generation of a B+tree-based index structure, search time is getting sub-linear. First, the cloud server is assumed to be “Honest-but-Curious”. Second, it is considered that the data are stored on an untrusted cloud server. The Merkle B-tree (MB-tree) is utilized to verify search results in the presence of access control rules when the cloud server is untrusted. Experiments are conducted using the real world dataset to confirm efficiency and precision of the offered scheme.

Technical Perspective

One of the most important functionalities of a database system is to answer queries. We are interested in the following question: If there exists more than one answer to the given query, which one should the database report? There are two apparent choices: to return all the valid answers or to return one of them. The problem with the former choice is that it is often time-prohibitive to search for all valid answers. In the latter choice, fairness may become an issue, since the index built for fast search may introduce bias to the query result. For example, the index may favor a certain portion of the input data (e.g., nodes near the root of a tree index) and with a higher chance, output an answer related to that portion than other portions. Such bias can sometimes lead to undesirable consequences.

Sombor index of trees with at most three branch vertices

Let G be a graph with set of vertices V(G) and set of edges E(G). The Sombor index is a vertex-degree-based-topological index recently introduced by Ivan Gutman, defined asSO(G)=∑uv∈E(G)(du)2+(dv)2.In this paper we determine the extremal values of SO over trees with at most three branch vertices.

More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons

AbstractLet G be a connected graph with vertex set V(G) and edge set E(G). The Sombor index of G is defined as and the reduced Sombor index of G is defined as where du denotes the degree of vertex u in G. In this paper, we determine maximum and minimum (reduced) Sombor index of chemical trees with given pendent vertices, and characterize their extremal graphs. In addition, some numerical results are discussed. We calculate the (reduced) Sombor index of a set of benzenoid hydrocarbons. The regression models show that boiling points and (reduced) Sombor index of benzenoid hydrocarbons are highly correlated.

Fast and Secure Location-Based Services in Smart Cities on Outsourced Data

With the advancement of mobile internet, cloud computing, and smart sensing devices, Location-Based Services (LBSs) have become more and more indispensable in the Internet of Things (IoT) based smart cities. Especially, spatial keyword queries have been widely deployed in real-life applications in recent years. Recently, several privacy-preserving spatial keyword queries schemes were proposed to guarantee data security and query privacy on outsourced data. However, these schemes support neither dynamic update nor diverse query types, which cannot meet the requirements in practical applications. This paper proposes two Secure Dynamic Spatial Keyword Queries (SDSKQ) constructions that support expressive query types and dynamic update. Firstly, we present a basic SDSKQ construction based on hidden-vector encryption and order-revealing encryption. Specifically, we propose a secure hybrid index structure for spatio-textual data, named Encrypted Textual Signature Quadtree (ETSQ-tree). Using ETSQ-tree, the server can prune the index tree according to search queries to reduce the search space. Besides, the ETSQ-tree can be updated dynamically. To resist the file-injection attack, which aims to infer query information according to newly inserted objects, we further improve the basic SDSKQ to achieve forward security. We implement our two constructions and evaluate them using real-world datasets. The experimental results show that they are efficient and feasible in practical applications, and the comparative evaluation confirms that the performance of our constructions outperforms that of the state-of-the-art schemes.

Entire Harary index of graphs

The Harary index H(G) of a connected graph G is defined as the sum of reciprocals of distances between all pairs of vertices in G [9, 13]. In this paper, as a continuing the study of entire indices like entire Zagreb indices of graphs [1], we introduce the entire Harary index by adding the reciprocals of distances between all pairs of edges and all pairs of edges and vertices to the Harary index. Our motivation in doing this coming from the following fact about molecular graphs: The intermolecular forces do not exist only between the atoms, but also between the atoms and bonds, so the relations between the atoms and bonds and also between the bonds together should be taken in account. Exact values of some families of graphs and some graphs operations are obtained. Simple formula of the entire Harary index of trees and some bounds are established.

An Approach For Combining Spatial And Textual Skyline Querying Using Indexing Mechanism

Most research works focus on either static data query processing or spatial data querying processing due to its large-sized data and the complex challenges that occur during its exploration, which is imperative. There are many state-of-the-art spatial indexing mechanisms for skyline querying to support either point/static or spatial objects, an indexing mechanism for skyline querying that supports both textual and spatial object querying is explored in this paper using an R* tree indexing mechanism that inserts data faster than its variant R tree technique and takes minimum execution time than the traditional spatial indexing mechanisms. The textual data is indexed using inverted files and attached to the R* tree. The proposed algorithm also works competitively under dynamic databases.

The Sombor Index of Trees and Unicyclic Graphs with Given Maximum Degree

Let $d_G(v)$ be the degree of the vertex $v$ in a graph $G$. The Sombor index of $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, which is a new degree-based topological index introduced by Gutman. Let $\mathscr{T}_{n,\Delta}$ and $\mathscr{U}_{n,\Delta}$ be the set of trees and unicyclic graphs with $n$ vertices and maximum degree $\Delta$, respectively. In this paper, the tree and the unicyclic graph with minimum Sombor index among $\mathscr{T}_{n,\Delta}$ and $\mathscr{U}_{n,\Delta}$ are characterized.

Biodiversity index of fruit trees cultivated by communities around marginal land in Jombang Regency, East Java Province, Indonesia

Marginal land conditions are a major problem in reducing the level of diversity of species due to decreased soil, climate, and water quality and topography. This study aims to analyze the diversity index and the dominance of fruit tree species cultivated by surrounding communities. The research method uses nested sampling in 4 (four) sub-districts, each of which is taken three sample villages and each of them is made of three transect plots measuring sample plots 20 × 20 m 2 (trees), 10 × 10 m 2 (poles), and 5×5m 2 (sapling). Data analysis using the Biodiversity R application program with version R 3.6.1. The results of the analysis of the growth rates of the best tree species were found in Ploso Regency because they had the highest growth rates of trees, poles, and saplings. The diversity of fruit tree species is based on the distribution of the highest H index value of 1.47 and the lowest of 0.58 and includes areas that have an abundance of moderate species. Fruit tree species that have the highest importance value index are Mangifera indica (mango) 70.26% with a tree density of 139 individuals per hectare, then Dimocarpus longan (longan) 40.24% with a tree density of 153 individuals per hectare. The results of this study indicate that the community structure of fruit trees in degraded lands is more dominated by plants from the Sapindaceae and Anacardiaceae families which generally grow spread in the 4 sub-districts studied.

Practical approximate indoor nearest neighbour locating with crowdsourced RSSIs

In the indoor space, finding the nearest neighbour is of great importance in location-based services. Received Signal Strength Indication (RSSI) has received much attention due to its simplicity and compatibility with existing hardware, which has been widely used for indoor localization. Existing indoor nearest neighbour search methods are based on the real walking distance, which need ground survey and much labor work to measure many real distances. Crowdsourcing is a low-cost and efficient way to collect the RSSI of indoor space without expert surveyors and designated coordinates for RSSI collection points. The crowdsourced RSSIs can reflect the location of indoor objects and RSSI-based localization method is the simplistic method as it needs low hardware requirements, low deployment cost and no survey indoor distance. So we study how to search the nearest neighbour of indoor objects with crowdsourced RSSIs. To address this problem, we propose a graph with interval weights, called I-graph, which can connect the RSSIs and represent the topology of indoor space. We also construct a search tree index D-tree, which can index the graph with interval weights and search the nearest neighbour objects efficiently. We also propose a novel distance metric for RSSI and study the relationship between the RSSI distance and the indoor distance. To locate nearest neighbour of indoor objects with crowdsourced RSSIs, we devise efficient search algorithms and pruning strategies for computing the nearest neighbour query. We demonstrate the efficiency and effectiveness of the proposed solution through extensive experiments with two real data sets.

Secure and Verifiable Keyword Search in Multiple Clouds

Searchable encryption enables users to perform a keyword-based search over encrypted data. Most of the existing searchable encryption schemes are based on the single cloud server model, which has some limitations, such as the single point of failure and malicious insiders. In contrast, the multicloud server model has higher data availability and security. However, directly moving the existing keyword search schemes to the multicloud would cause larger storage and search costs, and the data security cannot be distinctly improved. In addition, different from single cloud, a challenge is how to effectively verify query integrity when some cloud servers return incorrect or incomplete results in a multicloud situation. To ensure the data security, we propose an iterative encryption method to ensure the file privacy even if multiple cloud servers collude with each other. We use Bloom filter to map keywords and construct a tree index for each cloud server to achieve secure and efficient distributed keyword search. To verify the query integrity, we propose a fast verification algorithm to embed the file content and index information into the file ID. The analysis and experiments demonstrate the security and effectiveness of our scheme.