Wiener Index Research Articles

Degree and distance based topological descriptors of power graphs of finite non-abelian groups

A topological descriptor is a numerical value derived from the molecular structure that encapsulates the most important structural characteristics of the molecule under consideration. Fundamentally, it involves assigning an algebraic value to the composition of chemicals while developing a relationship between this value and several physical properties, like biological activity, and chemical reactivity. This article examines multiple kinds of degree and eccentricity-based topological indices for power graphs of various finite groups. We calculate the Wiener index and its reciprocal, atom-bond connectivity index and its fourth version, the Schultz index, the geometric–arithmetic and harmonic indices, and finally determine the general Randić and Harary indices of power graphs of finite cyclic and non-cyclic groups of order pq, dihedral, and generalized quaternion groups, where p,q(q≥p) are distinct primes.

Plant-Soil Carbon Storage in Dynamic Succession of Ecological Restoration in National Grassland Natural Park

Ecological restoration has a positive impact on global climate change. How plant-soil stores carbon in degraded grassland ecological restoration requires long-term monitoring and support. To reveal the dynamics of plant-soil carbon storage in the succession process of ecological restoration, compare the effects of artificial interference and natural restoration, and determine the impact of climate change and biodiversity on vegetation soil carbon storage, we conducted a study in National Grassland Natural Park, which is located on the southern foot of the Yinshan Mountains in Hohhot, Inner Mongolia, China. Based on long restoration chronosequences (2012–2022), using a space-for-time substitution approach and one-way ANOVA tests, Pearson correlation and structural equation modeling were used to investigate the interactions among these various factors. The results indicated that the carbon storage of aboveground vegetation first increased, and then, decreased with time. The underground root carbon storage and soil carbon storage at 0–10 cm and 20–30 cm first increased, then decreased, and finally, stabilized. The highest soil carbon storage (0–30 cm) was 102.11 t/ha in 2013, which accounted for 96.61% of the total organic carbon storage. The Shannon–Wiener index, individual number of species, and surface root carbon storage (0–10 cm) significantly increased the carbon storage of surface soil (0–10 cm) (p < 0.05). Compared to natural restoration, artificial restoration over seven years decreased soil carbon storage at 0–30 cm and underground root carbon storage at 0–10 cm (p < 0.05). Consequently, combining artificial restoration with natural restoration can help in establishing a more stable ecosystem faster and in increasing the carbon storage of the ecosystem. It is an effective management measure to promote grassland restoration in arid areas. Also, climate (MAT, MAP) change was closely correlated with plant-soil carbon storage.

Minimum Wiener index of unicyclic chemical graphs with girth 3

Minimum Wiener index of unicyclic chemical graphs with girth 3

Phytodiversity of Herbaceous Flora in the Grasslands of Jannaram Division, Kawal Tiger Reserve, Telangana, India

Aims: To study the diversity and composition of herbaceous species in the grasslands of Jannaram division, Kawal Tiger Reserve, Telangana, India.
 Place and Duration of Study: Grasslands of Jannaram division, Kawal Tiger Reserve, Telangana, July 2022 to August 2023.
 Methodology: The study was carried out by laying out 223 sample plots in the Jannaram division by dividing Jannaram division into 3 ranges. The plot size was 1 x 1 m. Varied diversity indices such as Simpson, Shannon–Weiner, species evenness, and IVI were calculated according to standard formulae. Vegetation analysis and IVI value of each species were calculated and analysed.
 Results: A total of 65 species, comprising 53 genera and 21 families, were reported. The dominant family in the study area was Poaceae, followed by Fabaceae, Malvaceae, Acanthaceae, Asteraceae and Euphorbiaceae. The Shannon–Weiner index was 3.46 and the Simpson index was 0.05, indicating high diversity with less dominance of herb layer in the study area. Top five species holding highest IVI value are Iseilema laxum (29.94), Cynodon barberi (12.19), Eragrostis amabilis (12.15), Dichanthium annulatum (11.20), and Apluda mutica (10.21).
 Conclusions: This research produced fundamental information regarding the types of herbaceous plants and grasses present at the Jannaram, Kawal Tiger Reserve. This would serve as a guide to the systematists, ethnobotanists, amongst several others researchers who are interested in identifying.

Passive restoration of fynbos after afforestation with exotic pines, South Africa

Plantations, and associated invasions, of exotic Pinus trees occur extensively in the southern hemisphere, threatening the persistence of biodiverse Mediterranean‐climate vegetation. Large‐scale decommissioning of such plantations in the eastern Cape Floristic Region (South Africa) enabled a wide‐ranging study showing successful passive restoration of fynbos vegetation after afforestation. Using a paired study design, we compared the diversity, and floristic and growth‐form composition of post‐fire recovering fynbos in former plantations with that in recently burnt neighboring fynbos in a natural state within the Garden Route National Park. The fynbos of the study area showed good autogenic recovery after several decades of pine afforestation and a fire subsequent to the clearcutting of these trees. On average, native plant abundance and species richness (particularly of shrubs) were significantly lower, and non‐native plant species richness significantly higher, in former plantation areas than in natural fynbos, but these differences were small. Species diversity (Shannon–Wiener index) did not differ significantly between the two vegetation states. The former plantations harbored 91% of the number of native species that the natural fynbos had, while the similarity of the two floras was 65%. Non‐metric multidimensional scaling ordination and multivariate generalized linear models accordingly showed no clear distinction in community composition between the two vegetation states. We concluded that the restoration potential of the montane grassy fynbos in the study area is superior to that previously documented in montane proteoid fynbos, and that fire and invasive alien plant control after clearcutting of the plantations is essential to promote fynbos restoration.

TOPOLOGY INDEX OF THE COPRIME GRAPH FOR DIHEDRAL GROUP OF PRIME POWER ORDER

In the field of molecular chemistry, graph theory is utilized to represent the structure of a molecule, where the set of nodes corresponds to its chemical elements and the set of edges represents the bonds within the chemical molecule. Graph theory, a mathematical discipline, finds application in various domains, one of which is group representation. This research will delve into the topic of the topological indices of the coprime graph of dihedral groups. The methodology employed involves reviewing several references related to dihedral groups, coprime graphs, and topological indices. This study yields results in the form of Harmonic index, Harary index, first Zagreb index, Gutman index, and Wiener index.

Significant Dynamic Disturbance of Water Environment Quality in Urban Rivers Flowing through Industrial Areas

The urban water environment is seriously affected by human activities. Rivers in highly industrialized areas, which often carry various types of industrial pollutants, such as metals and nutrients, are especially affected. In this study, the water quality of the Pi River, an industrial base that flows through Chengdu, a large city in Southwest China, was tested for one year. Heavy metal concentrations in the water, sediment, and macrobenthic and algal communities in the river were examined. The water pollution index (WPI) and trophic level index (TLI) were employed to measure the water pollution degree and eutrophication status, respectively. The Shannon—Wiener index (H’) and Margalef’s index (dM) were determined to represent the diversity and richness of macroinvertebrates. The principal component analysis (PCA) was employed to define the main heavy metal influencing factors in the Pi River. Our study showed that the eutrophication status increased with spatial change, and the eutrophication status was the most serious in the downstream reach, which was moderately eutrophic. The water body of the Pi River was seriously polluted by heavy metals, and the content of chromium (Cr) in the sediment and cadmium (Cd) in the water/sediment was far beyond the prescribed limit. In addition, we found that Cd had a serious impact on both the benthic and algal communities, and the benthic community structure was completely changed, destroying the original aquatic environment. We explored the mechanisms of the influence of Cd on aquatic fauna, and this information is of great significance for the future conservation of industrial urban rivers. In this study, the spatial–temporal variations in water quality and aquatic communities revealed the pollution status of a river flowing through industrial areas, which provided a basis for future river conservation and restoration.

Current Status and Potential Invasiveness Evaluation of an Exotic Mangrove Species, Laguncularia racemosa (L.) C.F. Gaertn, on Hainan Island, China

Laguncularia racemosa is widely planted in coastal mudflats as a pioneer species of mangrove afforestation in China; however, whether it exhibits invasiveness as an exotic species has been a topic of debate. In this study, we investigated the distribution, community structure, and population characteristics of L. racemosa on Hainan Island, China, to discuss its potential invasiveness. The results showed that Hainan Island has become an important distribution area for L. racemosa in China, covering a total area of 64.90 ha. In this investigation, a total of 11 true mangrove species, belonging to six families and 11 genera, were discovered in the L. racemosa community throughout Hainan Island. Furthermore, L. racemosa exhibited a strong adaptive range to seawater salinity and demonstrated a high capacity for natural dispersion. L. racemosa exhibited high importance values in the tree, shrub, and seedling layers of this artificial community, indicating its dominance and potential exclusion of other mangrove species. Over time, the importance values of L. racemosa in the tree and shrub layers initially decreased and then increased, while the opposite trend was observed in the seedling layer. Overall, the diversity indexes, including the Simpson index, Shannon–Wiener index, and Pielou community evenness index, were low in the planted L. racemosa forest on Hainan Island, suggesting a relatively simple species composition within the L. racemosa community. Additionally, based on the height levels of the L. racemosa population, we found that it mainly belonged to the first level, characterized by a large number of seedlings on Hainan Island. The number of L. racemosa seedlings gradually decreased in subsequent levels, and higher height levels of L. racemosa appeared with longer planting times, indicating a growing trend of potential invasiveness of L. racemosa on Hainan Island over an extended period. We suggest that the government should be able to list L. racemosa as an invasive species that can be harvested in the future. Based on this, this study provides fundamental data and theoretical references for the sustainable prevention, control, and management of the exotic species L. racemosa on Hainan Island, and also in other mangrove wetlands in China.

On the Distance Spectrum and Distance-Based Topological Indices of Central Vertex-Edge Join of Three Graphs

In this paper, we introduce a new graph operation based on a central graph called central vertex-edge join (denoted by $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$) and then determine the distance spectrum of $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$ in terms of the adjacency spectra of regular graphs $G_1$, $G_2$ and $G_3$ when $G_1$ is triangle-free. As a consequence of this result, we construct new families of non-D-cospectral D-equienergetic graphs. Moreover, we determine bounds for the distance spectral radius and distance energy of the central vertex-edge join of three regular graphs. In addition, we provide its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, Wiener index. Using these results, we calculate the topological indices of the organic compounds Methylcyclobutane $(C_5H_{10})$ and Spirohexane $(C_6H_{10})$.

Plant Community and Soil Properties Regulate Space-Scale Dependence of Species Diversity under Grazing Exclusion and Rest Grazing in the Qilian Mountains of China

Grazing exclusion (GE) and rest grazing (RG) are important management systems for the restoration of degraded grassland ecosystems. In order to evaluate the effects and mechanisms of different grazing management systems on the scale dependency of species diversity, plant community indices and soil variables were determined in 32 plots in the Qilian Mountains in Gansu Province. The results show that diversity effects and their regulating mechanisms had space-scale dependence under different grazing management systems. The species richness and species diversity indices of RG grassland were significantly higher than those of GE at the regional scale. Species richness for RG and GE plots was 128 and 98, respectively, and the Shannon–Wiener and Simpson indices for RG and GE plots were 2.26 and 2.16, and 0.85 and 0.83, respectively. Additionally, three grazing management systems had a significant effect on species richness in mountain meadows, but different management systems had significantly different effects on species diversity indices in mountain meadows and temperate steppes. Meanwhile, soil variables only influenced species diversity at the regional scale. Most of the plant and soil variables at each scale had positive effects on species diversity. However, a negative correlation was seen between biodiversity and species coverage, mean plant height, soil porosity (SP) and bulk density (BD) under the two contrasting grazing management systems. In conclusion, choosing RG at the regional scale and selecting grazing management systems according to different grassland types at the local scale can help to restore degraded grassland vegetation.

Bacterial communities structure in constructed wetlands for municipal and industrial wastewater treatment in Tanzania

Abstract Bacteria are the largest and most essential microorganisms in wastewater treatment systems. Despite the vital role bacteria play in enhancing wastewater treatment, its community structure and diversity remain poorly known. This study elucidated bacterial diversity and community structure in constructed wetlands (CWs) for municipal and industrial wastewater treatment using a culture-independent approach. Analysis of physicochemical parameters and Illumina high-throughput sequencing of V3 and V4 hypervariable regions of 16S rRNA gene bacterial community was performed. The results showed that the Proteobacteria were dominant (48.66%) phyla across all CWs. The Gammaproteobacteria class (27.67%), family Comamonadaceae (35.79) and Genus Flavobacterium (4.35%) were dominant across studied CWs. Bacterial abundance increased from the inlet to the outlet, with CWs having a higher abundance around the outlet, showing a good performance. The Shannon–Wiener index showed the highest species diversity (H = 1.45) in Banana Investment Limited CWs treating industrial wastewater. This study found significant bacterial diversity across studied CWs, with higher abundance and diversity at the outlet indicating better performance than those with low abundance and diversity at the outlet. We recommend a longitudinal investigation at each CWs section to confirm and establish the role of bacterial diversity on the performance pattern of CWs.

Wiener index and graph energy of zero divisor graph for commutative rings

Let [Formula: see text] be commutative ring and [Formula: see text] be the set of all non-zero zero divisors of [Formula: see text]. Then [Formula: see text] is said to be the zero divisor graph if and only if [Formula: see text] where [Formula: see text] and [Formula: see text]. Graph energy [Formula: see text] is defined as the sum of the absolute eigenvalues of the adjacency matrix [Formula: see text], then [Formula: see text]. Wiener index [Formula: see text] is defined as the sum of all distance between pairs of vertices [Formula: see text] and [Formula: see text], then [Formula: see text]. In this paper, we compute the graph energy from the adjacency matrix of the zero divisor graph and the Wiener index from the zero divisor graph associated with commutative rings.

Wiener index of an ideal-based zero-divisor graph of commutative ring with unity

The Wiener index of a connected graph G is W ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u , v ) . In this paper, we obtain the Wiener index of H-generalized join of graphs G 1 , G 2 , … , G k . As a consequence, we obtain some earlier known results in [Alaeiyan et al. in Aust. J. Basic Appl. Sci. (2011) 5(12): 145–152; Yeh et al. in Discrete Math. (1994) 135: 359–365] and we also obtain the Wiener index of the generalized corona product of graphs. We further show that the ideal-based zero-divisor graph Γ I ( R ) is a H-generalized join of complete graphs and totally disconnected graphs. As a result, we find the Wiener index of the ideal-based zero-divisor graph Γ I ( R ) and we deduce some of the main results in [Selvakumar et al. in Discrete Appl. Math. (2022) 311: 72–84]. Moreover, we show that W ( Γ I ( Z n ) ) is a quadratic polynomial in n, where Z n is the ring of integers modulo n and we calculate the exact value of the Wiener index of Γ Nil ( R ) ( R ) , where Nil(R) is nilradical of R. Furthermore, we give a Python program for computing the Wiener index of Γ I ( Z n ) if I is an ideal of Z n generated by pr , where pr is a proper divisor of n, p is a prime number and r is a positive integer with r ≥ 2 .

Some empirical and theoretical attributes of random multi-hooking networks

Multi-hooking networks are a broad class of random hooking networks introduced in [H. Mahmoud, Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment, Adv. Appl. Math. 111 (2019), p. 101930.] wherein multiple copies of a seed are hooked at each step, and the number of copies follows a predetermined building sequence of numbers. For motivation, we provide two examples: one from chemistry and one from electrical engineering. We explore the empirical and theoretical local degree distribution of a specific node during its temporal evolution. We ask what will happen to the degree of a specific node at step n that first appeared in the network at step j. We conducted an experimental study to identify some cases with Gaussian asymptotic distributions, which we then proved. Additionally, we investigate the distance in the network through the lens of the average Wiener index for which we obtain a theoretical result for any building sequence and explore its empirical distribution for certain classes of building sequences that have systematic growth.

Wiener index and vulnerability parameters of graphs

Let G be a finite simple graph with vertex set V(G). We define the Wiener index of G as W(G)=∑{u,v}∈V(G)d(u,v), where d(u,v) is the distance between u and v. This topological index was defined by Wiener in 1947 for studying the structural graphs of molecules (Wiener, 1947 [17]) (although, Wiener’s definition included a 1/2 multiple in front of the sum). Since that time, many theoretical results have been derived. The results we present concern the graph parameters integrity, toughness, tenacity and binding number. Lower bounds on these parameters have been used extensively to determine properties of graphs. For each of the four parameters, we provide a best possible upper bound on the Wiener index of G that guarantees a specified lower bound on that parameter. This provides a continuation of results provided in Feng et al. (2017).

ON THE GIRTH, INDEPENDENCE NUMBER, AND WIENER INDEX OF COPRIME GRAPH OF DIHEDRAL GROUP

The coprime graph of a finite group , denoted by , is a graph with vertex set such that two distinct vertices and are adjacent if and only if their orders are coprime, i.e., where |x| is the order of x. In this paper, we complete the form of the coprime graph of a dihedral group that was given by previous research and it has been proved that if , for some and if . Moreover, we prove that if is even, then the independence number of is , where is the greatest odd divisor of and if is odd, then the independence number of is . Furthermore, the Wiener index of coprime graph of dihedral group has been stated here.

Differences in Brain Functional Network Topology in High and Low Working Memory Performance

Nowadays the network approach in neuroscience provides a promising way of analyzing neurophysiological mechanisms that underlie psychological functions and is widely used to study working memory. To date, data obtained in neuroimaging studies links working memory with topological features of brain networks, such as increased connectivity between frontal, parietal, and temporal regions, as well as increased integration in brain networks as a whole. The present study examines the relationship between the topological characteristics of functional brain networks with the performance in the Sternberg item recognition paradigm based on electroencephalographic data. It is shown that the higher performance in Sternberg paradigm, implying a higher efficiency of the processes of encoding, storage, and retrieval of information from working memory, is associated with an increase in the integration of functional networks, expressed in differences in the clustering coefficient, participation coefficient, Wiener index and eigenvector centrality between the groups of high and low working memory task performance (p < .01). In addition, our data suggest the variability in the topological pattern of connectivity, which can be traced through changes in the magnitude of the standard deviation of the values of topological metrics during the task.

Characteristics and Relationships between Species Diversity and Productivity of Different Grassland Types in the Burqin Forest Region of the Altai Mountains

The Altai Mountain is located at the intersection of cold and arid regions. Climate change and overgrazing directly affect the growth of the grassland ecosystem in this region. This study took the grassland community in the Burqin forest area of the Altai Mountains as an example to analyze the species diversity and productivity of different grassland types based on 50 sampling plots. The relationship between species diversity and the productivity of grassland types at different altitudes was also discussed. The results showed that: (1) In the Burqin forest area, the desert steppe was dominated by Gramineae, Compositae, and Leguminosae. Montane steppe and mountain meadows were dominated by Umbelliferae, Gramineae, and Liliaceae. Gramineae, Ranunculaceae, and Liliaceae were the dominant families in both montane meadows steppe and alpine steppe but occurred in varying proportions. (2) The variation in plant numbers at the family–species level was characterized as montane steppe > alpine meadow > montane meadow steppe > desert steppe > mountain meadows. (3) The similarity in species composition among communities of different grassland types was low. In DS and MM, the number of grasslands and species played a decisive role in determining the biomass. The biomass of MS and AM was determined using the Simpson and Shannon–Wiener indices, and the number of dominant species determined the biomass. The number of grasslands and species did not have a significant impact on the biomass of MMS, which may be due to human factors such as grazing. (4) The dominant species in the grassland at an altitude of 1200–1400 m is prominent. The number of individuals in the dominant species was large and evenly distributed. Margalef peaked at an altitude of 1600–1800 m, but the number of individuals was small, resulting in low biomass and diversity in this range. Most plants survive at an altitude of 2000–2200 m; therefore, in grassland protection and planning management, it is important to consider the specific situation of plant growth under different habitats at different altitudes and make reasonable protection decisions based on local conditions to maintain species diversity and sustainable development of grassland ecosystems. This study provides basic data to support the theoretical basis for the protection and sustainable utilization of grassland resources and the restoration of degraded grasslands in the study region.

Soil properties and anthropogenic influences control the distribution of soil organic carbon in grasslands of northern China

AbstractSoil organic carbon (SOC) in grasslands not only plays an important role in carbon cycle but also largely affects soil function and thus determines grasslands' productivity. However, the influence mechanism of different factors on SOC storage in grasslands of northern China remains unclear. Based on an extensive field investigation and multiple statistical approaches, we mapped SOC of the grassland in northern China, and explored their correlations with climatic factors, biological factors, human activity intensity (HAI), and soil physicochemical factors. The effects of these factors on SOC were also analyzed by structural equation model. The results showed that the SOC varied across the regions and was strongly correlated with soil chemical factors (i.e., soil total nitrogen [TN], amorphous Al, and amorphous Fe) and soil physical factors (i.e., clay, silt, and sand). HAI, which was mainly influenced by grazing, had a significant but negative impact on SOC. Different from previous studies, we found that climatic factors such as temperature and precipitation indirectly affected SOC by influencing biological factors such as vegetation coverage, vegetation height, Shannon–Weiner index, and species richness. The soil physicochemical properties and HAI were the dominant influencing factors for controlling SOC distribution in the grasslands of northern China. Our findings will further increase the understanding of the carbon cycle in grassland ecosystems, while providing important scientific references for temperate grassland ecosystem management.

On the Wiener index of orientations of graphs

On the Wiener index of orientations of graphs