Wiener Index Research Articles

Graphs with a given conditional diameter that maximize the Wiener index

<abstract><p>The Wiener index $ W(G) $ of a graph $ G $ is one of the most well-known topological indices, which is defined as the sum of distances between all pairs of vertices of $ G $. The diameter $ D(G) $ of $ G $ is the maximum distance between all pairs of vertices of $ G $, and the conditional diameter $ D(G; s) $ is the maximum distance between all pairs of vertex subsets with cardinality $ s $ of $ G $. When $ s = 1 $, the conditional diameter $ D(G; s) $ is just the diameter $ D(G) $. The authors in <sup>[<xref ref-type="bibr" rid="b18">18</xref>]</sup> characterized the graphs with the maximum Wiener index among all graphs with diameter $ D(G) = n-c $, where $ 1\le c\le 4 $. In this paper, we will characterize the graphs with the maximum Wiener index among all graphs with conditional diameter $ D(G; s) = n-2s-c $ ($ -1\leq c\leq 1 $), which extends partial results above.</p></abstract>

Prediction of the Boiling Point of Acrylate and Methacrylate Polymers Through Wiener Index

Wiener index, a topological index of a molecular graph of the molecules are widely used to study the physico-chemical properties of the molecules. We considered the molecules, acrylate and methacrylate polymers. The Wiener index of these two polymers are obtained. And using this, the boiling point of these two polymers are studied extensively. Also, the detailed comparison is described graphically.

Topological indices of linear crossed phenylenes with respect to their Laplacian and normalized Laplacian spectrum

<abstract><p>As a powerful tool for describing and studying the properties of networks, the graph spectrum analyses and calculations have attracted substantial attention from the scientific community. Let $ C_{n} $ represent linear crossed phenylenes. Based on the Laplacian (normalized Laplacian, resp.) polynomial of $ C_{n} $, we first investigated the Laplacian (normalized Laplacian, resp) spectrum of $ C_{n} $ in this paper. Furthermore, the Kirchhoff index, multiplicative degree-Kirchhoff, index and complexity of $ C_{n} $ were obtained through the relationship between the roots and the coefficients of the characteristic polynomials. Finally, it was found that the Kirchhoff index and multiplicative degree-Kirchhoff index of $ C_{n} $ were approximately one quarter of their Wiener index and Gutman index, respectively.</p></abstract>

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.

Reduced plant species diversity and soil carbon and nitrogen contents driven by vegetation patchiness in alpine meadows

AbstractQuestionPatchiness of herbaceous species is a common feature of degraded alpine grasslands on the Qinghai–Tibet Plateau; however, the impact of this phenomenon on vegetation, soil seed bank (SSB), and soil carbon (C) and nitrogen (N) is largely unknown. We asked how different herbaceous patches affect above‐ground vegetation, the SSB and soil organic C (SOC) and total N (TN) contents in alpine meadows.LocationAn alpine meadow on the Qinghai–Tibetan Plateau, China.MethodsWe used field and greenhouse experiments to evaluate above‐ground vegetation parameters, the SSB composition, and SOC and TN contents of four commonly distributed herbaceous patches (Leymus secalinus, Kobresia humilis, Leontopodium nanum and Pedicularis kansuensis) and non‐patch vegetation (control).ResultsThere were significant differences in the species composition of above‐ground vegetation, with the plant species richness, Shannon–Wiener index, Simpson index, and Pielou evenness index reduced in the four patches compared to the control. In addition, patches had increased above‐ground biomass (AGB) and a lower below‐ground biomass to above‐ground biomass ratio (BGB:AGB); while the species diversity indices did not change considerably among the patches. The species composition of the SSB in the four patches differed from that in the control and the seed density increased markedly, but the species richness and diversity of the SSB remained unaltered. Moreover, the contents of SOC, total C, TN, microbial biomass C, and microbial biomass N in the four patches were markedly lower than in the control.ConclusionsThese findings suggest that patchiness of herbaceous species can alter the flora composition of alpine meadows and reduce plant species diversity, as well as the SOC and TN contents, which will adversely impact grassland biodiversity conservation and soil C sequestration, and then may intensify the degradation of alpine meadows on the Qinghai–Tibet Plateau.

Diversity of understory herbaceous plants during succession in rubber plantations in National Park of Hainan Tropical Rainforest, China

Abstract Revealing the impact of forest succession processes on changes in plant diversity is crucial for understanding the mechanisms that maintain plant diversity across various succession stages. While previous research has predominantly focused on the influence of environmental factors or management strategies on plant diversity within rubber plantation understories, there is a scarcity of studies examining the effects of forest succession processes on plant diversity. This study focuses on the plant diversity of the understory herbaceous layer within the rubber forest of the Yinggeling area, located in National Park of Hainan Tropical Rainforest. It employs a spatial analysis approach, rather than a temporal one, to examine the characteristics of the understory herbaceous community. The findings revealed that (1) The understory of Yinggeling rubber plantations harbors 175 plant species from 149 genera and 75 families, with Gramineae and Rubiaceae representing 46.45% of total species. And the dominant families are Rubiaceae, Gramineae, and Moraceae, with Ficus and Pteris being the dominant genera. (2) The dominant species vary with succession duration, with Tetrastigma pachyphyllum dominating in 0-year succession, Paspalum conjugatum in 3-year succession, and Microstegium fasciculatum in 7-year succession. (3) Diversity indices such as the Shannon–Wiener index, Simpson index, and Pielou index peak at 7 years of natural succession, while the species richness is highest at 3 years. (4) The similarity coefficient between understory herbaceous plant communities in rubber plantations undergoing 0 and 3 years of natural succession is highest 0.56, indicating a significant similarity, while similarity is lowest between 0 and 7 years of succession. This research shows that natural restoration helps increase species diversity in the understory herb layer of rubber forests. Succession leads to changes in the dominant families, genera, and species of the herbaceous layer. This change can be attributed to the intraspecific competition and ecological competition that occur during the succession process, leading to changes in biological and resource allocation.

Degree-weighted Wiener index of a graph

<abstract><p>From geometric point of view, we introduced the Sombor-Wiener index of a graph and studied the basic properties of the new index. It was shown that the Sombor-Wiener index was useful in predicting the acentric factor of octane isomers. In addition, we proposed a degree-weighted Wiener index to generalize the Schultz index, the Gutman index, and the Sombor-Wiener index. Meanwhile, we gave the calculation formula of degree-weighted Wiener index for generalized Bethe trees.</p></abstract>

A Survey of Recent Extremal Results on the Wiener Index of Trees

The Wiener index of a connected graph is defined as the sum of distances between all unordered pairs of its vertices. In this paper, we survey the known extremal results about the Wiener index of trees and the roots of the Wiener polynomials of trees from 2014 together with some open problems.

Habitat-dependent avian diversity in the agri-industrial zone of Jhajjar, Haryana, India

The study was performed to evaluate the habitat-dependent avian diversity in the southern region of Haryana from March 2021 to March 2023. The study aimed to identify avian species diversity in three habitats (agrifields, human settlement, and industrial zone) along with diversity indices and relative diversity. A total of 5037 birds belonging to 17 orders, 41 families, and 88 species were investigated with Simpson’s diversity index of 0.973 and Shannon– Wiener index of 4.04. Analysis of threat status revealed 38 species with stable population trends, 19 increasing, 20 decreasing, and 11 species having unknown trends. Out of 88 species, the IUCN status of two species Aquila nipalensis and Psittacula eupatria was endangered and near threatened, respectively; the rest 86 belonged to the least concern category. Of the total, 81% of species were residential, 10 were winter migratory, and 7 were summer migratory. Among the three habitats, agrifields were the most diverse, followed by human settlement and industrial zone. Diversified assemblages of birds suggested the region as a suitable habitat for both resident and migratory birds. This study highlights the need for more regular investigations to assess yearly avian diversity and possible threats to design comprehensive conservation plans.

On Some Topological Indices for the Orbit Graph of Dihedral Groups

Let ΓG be the orbit graph of G, with non-central orbits in the subset of order two commuting elements in G, and the vertices of ΓG connected if they are conjugate. The main objective of this study is to compute several topological indices for the orbit graph of a dihedral group, including the Wiener index, the Zagreb index, the Schultz index, and others. We also develop a relationship between the Wiener index and the other indices for the dihedral group’s orbit graph. Furthermore, their polynomial has been computed as well.

Study of Water Quality and Distribution of Phytoplankton in Hebbal Lake, Krishnarajanagara

Abstract: Life on earth depends on fresh water. Phytoplanktons are free floating, photosynthetic, aquatic microorganisms, which moves from one place to another. Phytoplanktons which are present in Hebbal Lake, Krishnarajanagara, and Mysore were studied with respect to their species diversity and distribution. The phytoplankton community is mainly represented by algal representatives including both prokaryotes and eukaryotic genera. Studies on phytoplankton and water quality of Hebbal Lake, Krishnarajanagar, Mysore Karnataka were undertaken for 5months from May 2023 to September 2023. The quantitative analysis of phytoplankton was done by Lackey’s drop method modified by saxena (1987).Diversity indices have been discussed by using PASTA Software Program. Physical and Chemical parameters were analysed using the standard methods of APHA (2025) and Trivedi and Gael (1984). Among the total 60 species recorded, 33species of Bacillariophyceae, 7species of Chlorophyceae, 8species of Cyanophyceae, 8 species of Desmidiaceae, 3species of Euglenophyceae are reported. Bacillariophyceae was the most dominant group followed by Chlorophyceae, Cyanophyceae, Desmidiaceae and Euglenophyceae. The result of physicochemical and biological parameters along with statistical method and biodiversity indices indicated that the Hebbal Lake is threatened ecologically due to various anthropogenic activities which lead organic pollution and eutrophication status of the lake. Statistical program Bray – Curtis similarity index explained, the cluster of EC, Total hardness and Total alkalinity is highly correlated with the clusters of phytoplanktons. Shannon and Weiner index showed that the lake was heavily polluted in the month of May due to increased temperature.

New indices to balance α-diversity against tree size inequality

The number and composition of species in a community can be quantified with α-diversity indices, including species richness (R), Simpson’s index (D), and the Shannon–Wiener index (H΄). In forest communities, there are large variations in tree size among species and individuals of the same species, which result in differences in ecological processes and ecosystem functions. However, tree size inequality (TSI) has been largely neglected in studies using the available diversity indices. The TSI in the diameter at breast height (DBH) data for each of 999 20 m × 20 m forest census quadrats was quantified using the Gini index (GI), a measure of the inequality of size distribution. The generalized performance equation was used to describe the rotated and right-shifted Lorenz curve of the cumulative proportion of DBH and the cumulative proportion of number of trees per quadrat. We also examined the relationships of α-diversity indices with the GI using correlation tests. The generalized performance equation effectively described the rotated and right-shifted Lorenz curve of DBH distributions, with most root-mean-square errors (990 out of 999 quadrats) being < 0.0030. There were significant positive correlations between each of three α-diversity indices (i.e., R, D, and H') and the GI. Nevertheless, the total abundance of trees in each quadrat did not significantly influence the GI. This means that the TSI increased with increasing species diversity. Thus, two new indices are proposed that can balance α-diversity against the extent of TSI in the community: (1 − GI) × D, and (1 − GI) × H'. These new indices were significantly correlated with the original D and H΄, and did not increase the extent of variation within each group of indices. This study presents a useful tool for quantifying both species diversity and the variation in tree sizes in forest communities, especially in the face of cumulative species loss under global climate change.

Using unmanned aerial vehicle multispectral data for monitoring the outcomes of ecological restoration in mining areas

AbstractThe effective and efficient monitoring of revegetation outcomes is a key component of ecosystem restoration. Monitoring often involves labor‐intensive manual methods, which are difficult to deploy when sites are inaccessible or involve large areas of revegetation. This study aimed to identify plant species and quantify α‐diversity index on a sub‐meter scale at Manlailiang Mine Site in Northwestern China using unmanned aerial vehicles (UAVs) as a means to semiautomate large‐scale vegetation monitoring. UAVs equipped with multispectral sensors were combined with three industry‐standard supervised classification algorithms (support vector machine [SVM], maximum likelihood, and artificial neural network) to classify plant species. Spectral vegetation indices (normalized difference vegetation index [NDVI], difference vegetation index [DVI], visible‐band difference vegetation index, soil‐adjusted vegetation index, modified soil‐adjusted vegetation index, and excess green–excess red) were used to assess vegetation diversity obtained from on‐ground survey plot data (the Margalef, Pielou, Simpson, and Shannon–Wiener indices). Our results showed that SVM outperformed other algorithms in species identification accuracy (overall accuracy of 84%). Significant relationships were observed between vegetation indices and diversity indices, with the DVI performing significantly better than many more commonly used indices such as the NDVI. The findings highlight the potential of combining UAV multispectral data, spectral vegetation indices and ground surveys for effective and efficient fine‐scale monitoring of vegetation diversity in the ecological restoration of mining areas. This has significant practical benefits for improving adaptive management of restoration through improved monitoring tools.

KARAKTERISTIK POHON INANG DAN KEANEKARAGAMAN ANGGREK (Orchidaceae) EPIFIT DI HUTAN LINDUNG LUNUNIANAM, DESA FATU’ULAN, KECAMATAN KI’E, KABUPATEN TIMOR TENGAH SELATAN, PROVINSI NUSA TENGGARA TIMUR

This study was conducted to determine the characteristics of orchid host trees, the diversity of epiphytic orchids (Orchidaceae), and the preservation of host trees and epiphytic orchids in the Lununianam protected forest area. This research was conducted in Lununianam Protection Forest, Fatu'ulan Village, Ki'e District, South Central Timor Regency, East Nusa Tenggara Province. Which is carried out for 1 month, from November to December 2022. This study used purposive sampling methods, surveys, and cruising paths with data analysis using qualitative descriptive and quantitative descriptive analysis. Host tree data was processed using vegetation analysis formulas and orchid data was processed using the Shannon wiener (H') diversity index formula.&#x0D; The results showed that there were 5 species of host trees found in the study site, namely forest guava (Syzygium pycnanthum), Matani (Pterocarpus indicus), Tahi (Celtis wightii planch), Kamala (Mallotus philippensis), Jenitri (Elaeocarpus ganitrid). Host trees found in Lununianam Protection forest have the characteristics of large hard and rough bark, have a height range of 12-30 m, and bark overgrown with moss, The results of vegetation analysis of host tree species with the highest Important Value Index namely Syzygium Pycnanthum and Elaeocarpus ganitrus have the lowest Important Value Index. Orchids found in the sampling plot in the Lununianam Protection Forest contained 8 species of epiphytic orchids, namely Dendrobium Sp, Vanda insignis, Eria multiflora, Flickingeria grandiflora, Trichoglottis bipenicillata, Pholidota carnea, Appendicula reflexa Blume, and Ceologyne speciosa. The calculation of Shannon Wiener's diversity index of 1.57 (H') is classified as medium (1&lt;H'&lt;3). The most common type of orchid found is the type Eria multifora with a total of 168 individuals and the type that is less commonly found is Appendicula reflexa Blume with 4 individuals.

Changes in Nutrient-Regulated Soil Microbial Communities in Soils Concomitant with Grassland Restoration in the Alpine Mining Region of the Qilian Mountains

In response to the significant ecological damage caused by unsustainable mining practices in the Qilian Mountains, ecological restoration projects have been undertaken in recent years. Analyzing the changes in soil microbial communities during the restoration process of mine meadows helps to reveal the mechanism of the restoration process in alpine mining areas. To explore the characteristics of soil microbial community distribution and their relationships with soil environmental factors during the restoration of alpine grasslands in the Qilian Mountains, we conducted surveys and analyses in two restoration levels low restoration (LR) and high restoration (HR) in the eastern Qilian Mountains, along with an undisturbed natural grassland control (NG). We found that as the degree of high-altitude mining area recovery increases, there were significant increases in vegetation cover, vegetation height, above-ground biomass, vegetation Shannon–Wiener index, soil organic carbon (SOC), soil water content (SWC), total nitrogen (TN), total phosphorus (TP), total potassium (TK), available phosphorus (AP), and available nitrogen (AN). Conversely, soil pH and electrical conductivity (EC) significantly decreased, with soil pH decreasing from 6.93 to 4.13. Restoration of high-altitude mining area grasslands significantly alters the distribution and composition of soil bacteria and fungi, while the impact on soil microbial community changes was not significant. Notably, with increasing recovery level, the dominant bacterial phyla are Acidobacteria and Proteobacteria, while the dominant fungal phyla are Ascomycota and Basidiomycota. These results indicate that changes in vegetation and soil properties both affect the composition of soil microbial communities, with soil properties having a greater influence. Soil fertility and nutrient levels emerge as the primary drivers influencing soil microbial composition communities and the degree of high-altitude mining area grassland recovery.

Impact of land use on macroinvertebrates from a multiscale perspective: enhancing structural equation models with inverse distance-weighted metrics

Effective riverine ecosystem management and restoration are predicated upon comprehending the relationship between land use patterns, the aquatic environment, and macroinvertebrates. However, the processes by which land use affects riverine ecosystems are exceedingly complex, influenced by factors across varying special scales. To explore these multi-scale relationships, we centered our study on the Taizi River Basin (TRB) in northeastern China, encompassing data on land use, macroinvertebrates, physical habitats, and water quality. We employed a synergy of correlation analysis, multiple stepwise regression, and redundant analysis techniques to discern these relationships at diverse spatial extents. This approach led to the development of 24 structural equation models (SEMs) that delineate the pathways bridging land use patterns to various biological indices. Remarkably, the interpretive capacity of SEM-Benthic Index of Biotic Integrity (SEM-BIBI), SEM-Shannon Wiener index (SEM-H’), and SEM-Average Score Per Taxon (SEM-ASPT) models was 76%, 70%, and 66% respectively. Within the TRB, the primary pathway from land use pattern to biological index consisted of: land use (predominantly rural and agricultural) feeding into physical habitat (habitat complexity, combined characteristic of speed and depth), which in turn affected benthic indicators. A secondary pathway charts a course from land use, through water quality (electrical conductivity and total dissolved solids) and culminating in benthic variations. Furthermore, our findings indicated that inverse-distance-weighted (IDW) metrics, specifically the IDW distance based on flow length to the stream outlet and the nearest stream (iFLO_m and iFLS_m, respectively) could provide more accurate interpretations of the river’s ecological health status. Given the effects of rural areas and agriculture on macroinvertebrates, we recommend establishing 100-m wide riparian buffers. The optimal buffer range should be determined by evaluating various biological needs and considerations.

BRYOPHYTE DIVERSITY AND ATMOSPHERIC POLLUTION IN A RESIDENTIAL AREA AND AN INDUSTRIAL URBAN FOREST IN JAKARTA, INDONESIA

Bryophytes are nonvascular plants that have simple structures that are sensitive to environmental changes, and they can, therefore be used as indicators of air quality. The presence of bryophytes in disturbed urban ecosystems, such as residential and industrial areas, indicates that their structures have adapted to survive in such areas. The objective of this study was to compare the bryophyte diversity and air quality indices between a residential area and an industrial area in Jakarta. The research was conducted in the Bona Indah residential area in South Jakarta and the Jakarta Industrial Estate Pulogadung (JEIP) urban forest. Sampling was carried out using the transect method in the residential area and the quadratic method in the urban forest on three different substrates, namely rocks or concrete, soil and tree trunks. The percentage of the epiphytic bryophyte cover was measured using a 10 × 10 cm subplot. Voucher specimens were stored at the Herbarium UI DEP and Herbarium IPB. Twenty-one species of moss and three species of liverwort were found in the two locations. Bryophytes were found on all the substrates in the residential area, but in the urban forest, they were found only on tree trunks and rock/cement substrates. Based on the Shannon–Wiener Index, although both locations had moderate bryophyte diversity, the residential area’s bryophyte diversity was higher than that of the urban forest. The index of atmospheric purity in the residential area was 4.3, indicating a high level of pollution, and it was 0.3 in the urban forest, showing that it was also very polluted.

Bounds of Fibonacci-sum graph for some distancebased topological indices and an application on BFS algorithm

The aim of this study is to obtain the lower and upper bounds for the distance-based Wiener index, Hyper-Wiener index and Harary index of the Fibonacci-sum graph. Firstly, the shortest distance of the vertices in the Fibonacci-sum graph is characterized. Afterwards, the numbers of decision vertices and leaves are obtained by applying the BFS algorithm to the Fibonacci-sum graph.

The Wiener, hyper-Wiener, Harary and SK indices of the P(Z_{p^k·q^r}) power graph

The undirected $P(Z_n)$ power graph of a finite group of $Z_n$ is a connected graph, the set of vertices of which is $Z_n$. Here $\langle u, v\rangle \in P(Z_n)$ are two diverse adjacent vertices if and only if $u \ne v$ and $\langle v \rangle \subseteq \langle u \rangle$ or $\langle u \rangle \subseteq \langle v \rangle$. We will shortly name the undirected $P(Z_n)$ power graph as the power graph $P(Z_n)$. The Wiener, hyper-Wiener, Harary and SK indices of the $P(Z_n)$ power graph are in order as follows $$\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right), \ \frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,d\left( u,v \right)+\frac{1}{2}\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,{{d}^{2}}\left(u,v \right),$$ $$\underset{\left\{ u,v \right\}\subseteq V\left( G \right)}{\mathop \sum }\,\frac{1}{d\left(u,v \right)} \mbox{ and } \frac{1}{2}\underset{uv\in E\left( G \right)}{\mathop \sum }\,\left( {{d}_{u}}+{{d}_{v}} \right).$$ In this article we focus more on the indices of $P(Z_n)$ power graph by Wiener, hyper-Wiener, Harary and &lt;em&gt;SK&lt;/em&gt; the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and &lt;em&gt;SK&lt;/em&gt; indices of the power graph $P(Z_n)$ corresponding to the vertex $n = p^k \cdot q^r$. These are as follows: $p, q$ are distinct primes and $k, r$ are nonnegative integers.

Some results on the Wiener index related to the Šoltés problem of graphs

The Wiener index, W(G), of a connected graph G is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs G with large Rm(G)=|{v∈V(G)∣W(G)−W(G−v)=m∈Z}|/|V(G)|. It is shown that there is a graph G with Rm(G)>1/2 for any integer m≥0. In particular, there is a regular graph of even degree with this property for any odd m≥1. The proposed approach allows to construct new families of graphs G with R0(G)→1/2 when the order of G increases.