Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network.

The Laplacian spectrum significantly contributes the study of the structural features of non-regular networks. Actually, it emphasizes the interaction among the network eigenvalues and their structural properties. Let Pn(Pn') represent the pentagonal-derivation cylinder (Möbius) network. In this article, based on the decomposition techniques of the Laplacian characteristic polynomial, we initially determine that the Laplacian spectra of Pn contain the eigenvalues of matrices LR and LS. Furthermore, using the relationship among the coefficients and roots of these two matrices, explicit calculations of the Kirchhoff index and spanning trees of Pn are determined. The relationship between the Wiener and Kirchhoff indices of Pn is also established.

Einstein Model of a Graph to Characterize Protein Folded/Unfolded States.

The folded structures of proteins can be accurately predicted by deep learning algorithms from their amino-acid sequences. By contrast, in spite of decades of research studies, the prediction of folding pathways and the unfolded and misfolded states of proteins, which are intimately related to diseases, remains challenging. A two-state (folded/unfolded) description of protein folding dynamics hides the complexity of the unfolded and misfolded microstates. Here, we focus on the development of simplified order parameters to decipher the complexity of disordered protein structures. First, we show that any connected, undirected, and simple graph can be associated with a linear chain of atoms in thermal equilibrium. This analogy provides an interpretation of the usual topological descriptors of a graph, namely the Kirchhoff index and Randic resistance, in terms of effective force constants of a linear chain. We derive an exact relation between the Kirchhoff index and the average shortest path length for a linear graph and define the free energies of a graph using an Einstein model. Second, we represent the three-dimensional protein structures by connected, undirected, and simple graphs. As a proof of concept, we compute the topological descriptors and the graph free energies for an all-atom molecular dynamics trajectory of folding/unfolding events of the proteins Trp-cage and HP-36 and for the ensemble of experimental NMR models of Trp-cage. The present work shows that the local, nonlocal, and global force constants and free energies of a graph are promising tools to quantify unfolded/disordered protein states and folding/unfolding dynamics. In particular, they allow the detection of transient misfolded rigid states.

Dataset complementary prism networks.

The complementary prism of G, denoted by GG¯, is the graph obtained from the disjoint union of G and G¯ by adding edges between the corresponding vertices of G and G¯. Up to date, the progress of experimental research around complementary prisms is limited by the unavailability of publicly available instances that could be used to run extensive experiments and to compare the performance on different topological index solutions and its bounds. For this reason, we decided to make publicly available 435 instances of type GG¯ randomly generated, with increasing network size (from 12 to 1948 nodes). The dataset presents instances of Complementary Prism Networks suitable to measure the Wiener Index and Generalized Wiener Index and the value of these indices for these instances. In addition, are presented the value of some lower and upper bounds proposed in the literature for these indices and their error with respect to the value of the index.

Forgotten Topological and Wiener Indices of Prime Ideal Sum Graph of Zn.

BACKGROUND: Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. OBJECTIVE: Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. METHODS: For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. RESULTS: In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. CONCLUSION: In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.

Heterogeneous effects of energy consumption structure on ecological footprint.

Attention to environmental sustainability has increased among nations, especially after the Paris Agreement and COP26 of 2021. Considering that fossil fuel consumption is one of the main factors causing environmental degradation, altering the energy consumption patterns of nations toward clean energy can be a suitable solution. For this purpose, this study investigates the impact of energy consumption structure (ECS) on the ecological footprint from 1990 to 2017. This research includes three steps: First, the energy consumption structure is calculated using the Shannon-Wiener index. Second, from 64 countries with middle- and high-income levels, the club convergence method is used to identify countries with similar patterns in an ecological footprint over time. Third, using the method of moments quantile regression (MM-QR), we examined the effects of ECS in different quantiles. The results of club convergence show that the two groups of countries with 23 and 29 members have similar behavior over time. The results of the MM-QR model show that for club 1, the energy consumption structure in quantiles of 10th, 25th, and 50th has positive effects on the ecological footprint, while in 75th and 90th are negative. The results of club 2 indicate that the energy consumption structure has positive effects on the ecological footprint in quantiles 10th and 25th, but negative effects on 75th. Also, the results show that GDP, energy consumption, and population in both clubs have positive effects, and trade openness has negative effects on ecological footprint. Considering that the results indicate that changing the structure of energy consumption from fossil fuels to clean energies improves the environmental quality, so governments should use incentive policies and support packages for the development of clean energy and reduce the costs of installing renewable energy.

The Connection of the Generalized Robinson-Foulds Metric with Partial Wiener Indices.

In this work we propose the partial Wiener index as one possible measure of branching in phylogenetic evolutionary trees. We establish the connection between the generalized Robinson-Foulds (RF) metric for measuring the similarity of phylogenetic trees and partial Wiener indices by expressing the number of conflicting pairs of edges in the generalized RF metric in terms of partial Wiener indices. To do so we compute the minimum and maximum value of the partial Wiener index [Formula: see text], where [Formula: see text] is a binary rooted tree with root [Formula: see text] and [Formula: see text] leaves. Moreover, under the Yule probabilistic model, we show how to compute the expected value of [Formula: see text]. As a direct consequence, we give exact formulas for the upper bound and the expected number of conflicting pairs. By doing so we provide a better theoretical understanding of the computational complexity of the generalized RF metric.

The Wiener index, degree distance index and Gutman index of composite hypergraphs and sunflower hypergraphs.

Topological invariants are numerical parameters of graphs or hypergraphs that indicate its topology and are known as graph or hypergraph invariants. In this paper, topological indices of hypergraphs such as Wiener index, degree distance index and Gutman index are considered. A g-composite hypergraphs is a hypergraphs that is obtained by the union of g hypergraphs with every hypergraph has exactly one vertex in common. In this article, results of above said indices for g-composite hypergraphs, where g ≥ 2 , are calculated. Further these results are used to find the Wiener index, degree distance index and Gutman index of sunflower hypergraphs and linear uniform hyper-paths.

Investigating Unused Tools for the Animal Behavioral Diversity Toolkit.

Behavioral diversity is a commonly used tool used to quantify the richness and evenness of animal behaviors and assess the effect of variables that may impact an animal's quality of life. The indices used in behavioral diversity research, and the study subjects, have not been formally reviewed. This paper aims to identify which indices are being used in behavioral diversity research, and under which scenarios, and uncover novel indices from other disciplines that could be applied to behavioral diversity. To investigate the techniques and species investigated in behavioral diversity literature, a Web of Science literature search was conducted. Two methods: behavioral richness and the Shannon-Wiener index, were the most frequently used indices, whereas the Behavioral Variability index featured rarely. While a range of species appeared in the behavioral literature, mammals were the most frequently studied Class, whereas amphibians did not feature in any papers. There are several diversity indices which did not feature in behavioral diversity including Simpson's index, and Chao. Such indices could be used to better understand animal behavioral study outputs or be used to estimate the number of 'unobserved' behaviors that an animal may express. Future studies could therefore extend beyond the Shannon-Wiener and richness indices.

The wiener index of the zero-divisor graph for a new class of residue class rings.

The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo p s q t for all distinct primes p, q and s , t ∈ N . We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph Γ E (R) are derived for R = Z p s q t .

Socioeconomic determinants of crop diversity in Bule Hora Woreda, Southern Ethiopia.

Crop diversification on the farm is a useful approach, especially in developing countries, where agriculture is the primary source of income. Crop diversity management on the farm is critical for reducing poverty, increasing farm revenue, creating jobs, and ensuring long-term agricultural sustainability by maintaining biodiversity, soil, and water resources. Despite their relevance, several variables are currently affecting farmers' crop production decisions. The purpose of this research was to see how socioeconomic factors influence crop diversification. We chose randomly 84 sample household heads from four kebeles to collect socioeconomic and on-farm data. The Shannon-Wiener index (SWI) and crop species richness were used to assess crop diversity. A multiple stepwise linear regression model was used to evaluate the data. Crop diversity was positively and significantly related to household farm size, animal size and composition, annual income, and the location's altitudinal gradient. A lack of road infrastructure and market connections constrained farmers' crop diversification options. It's vital to connect distant areas with road networks and market ties to promote farm-level crop diversification.

Altitudinal patterns of species richness and flowering phenology in herbaceous community in Qilian Mountains of China.

In montane systems, there are normally significant spatial differences in vegetation community structure and ecological processes due to the complex topography. The study of such topographic effect can provide scientific basis for the prediction of vegetation dynamics. In this work, the effects of altitude and slope aspect on species richness and flowering phenology of herbaceous communities were investigated in Qilian Mountains, a typical mountainous region in arid climate zones of China. Our monitoring of 102 plots in 34 sites revealed that there were significant topographic effects on species richness and flowering phenology. Specifically, the results showed a spatial pattern that the average number of species in plots was slightly higher at middle altitudes, and was higher on shady than sunny slopes. In flowering phenology, the flowering onsets of low-altitude and sunny-slope communities are generally earlier than that of high-altitude and shady-slope communities, respectively, while the ending dates of flowering between slope aspects and between altitudes are relatively small. This topographic effect revealed the influences of temperature and soil moisture on community structure and flowering phenology, which is reflected in the inverse responses of species richness to temperature and soil water content, and the high sensitivity of flowering phenology to temperature. It can be inferred that under the conditions of climate warming and wetting in the future, the species diversity of herbaceous community may increase at high altitudes, and the flowering duration is likely to be further prolonged in Qilian Mountains.

Hosoya Polynomial for Subdivided Caterpillar Graphs.

BACKGROUND: Computing Hosoya polynomial for a graph associated with a chemical compound plays a vital role in the field of chemistry. From Hosoya polynomial, it is easy to compute the Weiner index(Weiner number) and Hyper Weiner index of the underlying molecular structure. The Wiener number enables the identifying of three basic features of molecular topology: branching, cyclicity, and centricity (or centrality) and their specific patterns, which are well reflected by the physicochemical properties of chemical compounds. Caterpillar trees are used in chemical graph theory to represent the structure of benzenoid hydrocarbons molecules. In this representation, one forms a caterpillar in which each edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the corresponding rings belong to a sequence of rings connected end-to-end in the structure. Due to the importance of Caterpillar trees, it is interesting to compute the Hosoya polynomial and the related indices. METHODS: The Hosoya polynomial of a graph G is defined as H(G;x) = Σd(G)K=0 d(G.k)xk. In order to compute the Hosoya polynomial, we need to find its coefficient d(G.k) which is the number of pairs of vertices of G which are at distance k. We classify the ordered pair of vertices which are at distance , 2 ≤ m ≤ (n + 1)k in the form of sets. Then finding the cardinality of these sets and adding them up will give us the value of coefficient d(G.m) . Finally, using the values of coefficients in the definition, we get the Hosoya polynomial of uniform subdivision of caterpillar graph. RESULT: In this work, we compute the closed formula of Hosoya polynomial for subdivided caterpillar trees. This helps us to compute the Weiner index and hyper-Weiner index of uniform subdivision of caterpillar graph. CONCLUSION: Caterpillar trees are among the important and general classes of trees. Thorn rods and thorn stars are the important subclasses of caterpillar trees. The idea of the present research article is to provide a road map to those researchers who are interested in studying the Hosoya polynomial for different trees.

Odiel River (SW Spain), a Singular Scenario Affected by Acid Mine Drainage (AMD): Graphical and Statistical Models to Assess Diatoms and Water Hydrogeochemistry Interactions.

The Odiel River (SW Spain) is one of the most cited rivers in the scientific literature due to its high pollution degree, generated by more than 80 sulphide mines' (mostly unrestored) contamination in the Iberian Pyritic Belt (IPB), that have been exploited for more than 5000 years. Along the river and its tributaries, the physico-chemical parameters and diatoms, from 15 sampling points, were analyzed in the laboratory. Physico-chemical parameters, water chemical analysis, together with richness and Shannon-Wiener indexes were integrated in a matrix. An initial graphical treatment allowed the definition and proposal of a functioning system model, as well as the establishment of cause-effect relationships between pollution and its effects on biota. Then, the proposed model was statistically validated by factor analysis. For acidic pH waters, high values of Eh, TDS, sulphate, ∑REE and ∑Ficklin were found, while diatomologic indicators took low values. Thus, factor analysis was a very effective tool for graphical treatment validation as well as for pollution-biota interaction models' formulation, governed by two factors: AMD processes and water balance suffered by the studied river. As a novelty, the cause-effect relationships between high barium concentration and low diversity and richness were demonstrated in the IPB, for the first time.

Neurotoxicity of Pesticides: The Roadmap for the Cubic Mode of Action.

Pesticides are used today on a planetary-wide scale. The rising need for substances with this biological activity due to an increasing consumption of agricultural and animal products and to the development of urban areas makes the chemical industry to constantly investigate new molecules or to improve the physicochemical characteristics, increase the biological activities and improve the toxicity profiles of the already known ones. Molecular databases are increasingly accessible for in vitro and in vivo bioavailability studies. In this context, structure-activity studies, by their in silico - in cerebro methods, are used to precede in vitro and in vivo studies in plants and experimental animals because they can indicate trends by statistical methods or biological activity models expressed as mathematical equations or graphical correlations, so a direction of study can be developed or another can be abandoned, saving financial resources, time and laboratory animals. Following this line of research the present paper reviews the Structure-Activity Relationship (SAR) studies and proposes a correlation between a topological connectivity index and the biological activity or toxicity made as a result of a study performed on 11 molecules of organophosphate compounds, randomly chosen, with a basic structure including a Phosphorus atom double bounded to an Oxygen atom or to a Sulfur one and having three other simple covalent bonds with two alkoxy (-methoxy or -ethoxy) groups and to another functional group different from the alkoxy groups. The molecules were packed on a cubic structure consisting of three adjacent cubes, respecting a principle of topological efficiency, that of occupying a minimal space in that cubic structure, a method that was called the Clef Method. The central topological index selected for correlation was the Wiener index, since it was possible this way to discuss different adjacencies between the nodes in the graphs corresponding to the organophosphate compounds molecules packed on the cubic structure; accordingly, "three dimensional" variants of these connectivity indices could be considered and further used for studying the qualitative-quantitative relationships for the specific molecule-enzyme interaction complexes, including correlation between the Wiener weights (nodal specific contributions to the total Wiener index of the molecular graph) and the biochemical reactivity of some of the atoms. Finally, when passing from SAR to Q(uantitative)-SAR studies, especially by the present advanced method of the cubic molecule (Clef Method) and its good assessment of the (neuro)toxicity of the studied molecules and of their inhibitory effect on the target enzyme - acetylcholinesterase, it can be seen that a predictability of the toxicity and activity of different analogue compounds can be ensured, facilitating the in vivo experiments or improving the usage of pesticides.

Edge Distance-based Topological Indices of Strength-weighted Graphs and their Application to Coronoid Systems, Carbon Nanocones and SiO2 Nanostructures.

The edge-Wiener index is conceived in analogous to the traditional Wiener index and it is defined as the sum of distances between all pairs of edges of a graph G. In the recent years, it has received considerable attention for determining the variations of its computation. Motivated by the method of computation of the traditional Wiener index based on canonical metric representation, we present the techniques to compute the edge-Wiener and vertex-edge-Wiener indices of G by dissecting the original graph G into smaller strength-weighted quotient graphs with respect to Djokovic-Winkler relation. These techniques have been applied to compute the exact analytic expressions for the edge-Wiener and vertex-edge-Wiener indices of coronoid systems, carbon nanocones and SiO2 nanostructures. In addition, we have reduced these techniques to the subdivision of partial cubes and applied to the circumcoronene series of benzenoid systems.

Genome hunting of carbonyl reductases from Candida glabrata for efficient preparation of chiral secondary alcohols.

In this work, genome hunting strategy was adopted in screening for reductases from Candida glabrata. A total of 37 putative reductases were successfully expressed in E. coli BL21(DE3). A substrate library containing 32 substrates was established for characterization of each reductase by average specific activity and Shannon-Wiener index. Among them, Cg26 was identified with the highest efficiency and wider substrate spectrum in the reduction of prochiral ketones, with average activity and Shannon-Wiener index of 8.95U·mg-1 and 2.82. Cg26 is a member of 'extended' short chain dehydrogenase/reductase superfamily. Ni2+ could improve its activity. As much as 150g·L-1 ethyl 2-oxo-4-phenylbutyrate could be completely converted by 10g·L-1 Cg26. This study provides evidence for this newly identified Cg26 in the preparation of chiral secondary alcohols.

Computational prediction of therapeutic peptides based on graph index.

As therapeutic peptides have been taken into consideration in disease therapy in recent years, many biologists spent time and labor to verify various functional peptides from a large number of peptide sequences. In order to reduce the workload and increase the efficiency of identification of functional proteins, we propose a sequence-based model, q-FP (functional peptide prediction based on the q-Wiener Index), capable of recognizing potentially functional proteins. We extract three types of features by mixing graphic representation and statistical indices based on the q-Wiener index and physicochemical properties of amino acids. Our support-vector-machine-based model achieves an accuracy of 96.71%, 93.34%, 98.40%, and 91.40% for anticancer, virulent, and allergenic proteins datasets, respectively, by using 5-fold cross validation.

Revisiting the Hunter Gaston discriminatory index: Note of caution and courses of change.

Hunter Gaston Discriminatory Index (HGDI) is a widely used estimator of discriminatory power of genotyping methods and diversity of molecular markers in bacterial pathogens, including Mycobacterium tuberculosis. In my opinion, the index is somewhat misleading: a closer look at common practice and particular studies reveals that values in the range of 0.6-0.9 are modest but uncritically perceived as high. I propose and discuss three courses of change: (i) to continue using HGDI but be aware of the true meaning behind its value and increase a threshold of acceptable resolution to the more adequate values of 0.90-0.99, depending on study design; (ii) to turn to other known indices of diversity (e.g., Shannon index), in order to complement HGDI; (iii) to develop new, intuitively more realistic estimator.

Distance-based topological polynomials and indices of friendship graphs.

Drugs and chemical compounds are often modeled as graphs in which the each vertex of the graph expresses an atom of molecule and covalent bounds between atoms are represented by the edges between their corresponding vertices. The topological indicators defined over this molecular graph have been shown to be strongly correlated to various chemical properties of the compounds. In this article, by means of graph structure analysis, we determine several distance based topological indices of friendship graph [Formula: see text] which is widely appeared in various classes of new nanomaterials, drugs and chemical compounds.

Wiener index on rows of unit cells of the face-centred cubic lattice.

The Wiener index of a connected graph, known as the `sum of distances', is the first topological index used in chemistry to sum the distances between all unordered pairs of vertices of a graph. The Wiener index, sometimes called the Wiener number, is one of the indices associated with a molecular graph that correlates physical and chemical properties of the molecule, and has been studied for various kinds of graphs. In this paper, the graphs of lines of unit cells of the face-centred cubic lattice are investigated. This lattice is one of the simplest, the most symmetric and the most usual, cubic crystal lattices. Its graphs contain face centres of the unit cells and other vertices, called cube vertices. Closed formulae are obtained to calculate the sum of shortest distances between pairs of cube vertices, between cube vertices and face centres and between pairs of face centres. Based on these formulae, their sum, the Wiener index of a face-centred cubic lattice with unit cells connected in a row graph, is computed.