2,970 Results Found

Topological indices and connectivity polynomials are invariants of molecular graphs. These invariants have the tendency of predicting the properties of the molecular structures. The honeycomb network structure is an important type of benzene network....

The recent discovery of boron triangular nanotubes competes with carbon in many respects. The closed form of M‐polynomial of nanotubes produces closed forms of many degree‐based topological indices which are numerical parameters of the structure and,...

The Sombor index has recently become a central tool among degree-based graph invariants; however, it does not explicitly isolate degree imbalance along edges. In this work, we introduce the degree-based Contrastive Sombor Index (CSO), which combines...

The Hyperbolic Sombor index $HSO\left(G\right)$ of a graph $G=\left(V\right(G),E(G\left)\right)$ is proposed as a new vertex-degree-based topological index. In this paper, we investigate the mathematical properties of this novel vertex-degree-based topological index for general graphs....

Consider a unicyclic graph G with edge set $E\left(G\right)$. Let $\mathfrak{f}$ be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as ${\mathcal{I}}_{\mathfrak{f}}($...

In this article, we introduce the neighborhood versions of two classical topological indices, namely neighborhood geometric–arithmetic and neighborhood atom bond connectivity indices. We study the graph-theoretic properties of these new topological i...

Titania is one of the most comprehensively studied nanostructures due to their widespread applications in the production of catalytic, gas sensing, and corrosion-resistant materials. M-polynomial of nanotubes has been vastly investigated, as it produ...

Dendrimers are highly branched organic macromolecules with successive layers of branch units surrounding a central core. The M-polynomial of nanotubes has been vastly investigated as it produces many degree-based topological indices. These indices ar...

The discovery of new nanomaterials adds new dimensions to industry, electronics, and pharmaceutical and biological therapeutics. In this article, we first find closed forms of M-polynomials of polyhex nanotubes. We also compute closed forms of variou...

The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topologi...

Many degree-based topological indices can be obtained from the closed-off M-polynomial of a carbon nanocone. These topological indices are numerical parameters that are associated with a structure and, in combination, determine the properties of the...

The phenomenon of urban sprawl has received much attention. Accurately confirming the spatial expansion degree of urban sprawl (SEDUS) is a prerequisite to controlling urban sprawl. However, there is no reliable metric to accurately measure SEDUS. In...

To address the critical issue of uneven flow allocation in district heating systems, this paper proposes a novel, systematic evaluation framework centered on the Flow Balance Index. The basic approach transforms discrete actual flow rates of users ac...

The concept of arithmetic-geometric index was recently introduced in chemical graph theory, but it has proven to be useful from both a theoretical and practical point of view. The aim of this paper is to obtain new bounds of the arithmetic-geometric...

The water–energy–food (WEF) nexus is central to regional sustainable development, particularly in resource-based regions (RBRs) facing resource depletion and ecological strain. While existing studies have examined internal WEF linkages an...

A vertex-degree-based (VDB, for short) topological index $\phi$ induced by the numbers $\left\{{\phi }_{ij}\right\}$ was recently defined for a digraph D, as ${\displaystyle \phi \left(D\right)=\frac{1}{2}\sum _{uv}{\phi }_{d_u^{+}{d}_v^{-}},}$ where $d_u^{+}$ denotes the out-degree of the vertex $u,$$d_v^{-}$ denotes the in-degree of the vertex $v,$ and the sum ru...

The large-scale integration of photovoltaic (PV) power can bring a greatly negative influence on the grid-connected system’s voltage stability. To study the static voltage stability (SVS) of PV grid-connected systems, the traditional SVS index,...

Several topological indices are known to have widespread implications in a variety of research areas. Over the years, the Kirchhoff index has turned out to be an extremely significant and efficient index. In this paper, we propose the exact formulas...

As Guangdong is a pivotal province in China’s national forest city initiative, examining the spatiotemporal evolution and key drivers of eco-environmental quality (EEQ) in Guangdong is essential for advancing regional sustainable development. T...

Realizing real-time and rapid monitoring of crop growth is crucial for providing an objective basis for agricultural production. To enhance the accuracy and comprehensiveness of monitoring winter wheat growth, comprehensive growth indicators are cons...

Vulnerability impacts have increased in an unprecedented way with the effects of global warming, climate change, erosion, sea level rise, tsunami, flood, and drought—natural events that jointly cause geomorphological changes, especially in coas...

Rapid urbanization often exerts massive pressure on the resources relied upon by the ecological environment. It is necessary to quickly evaluate the interaction and mutual influence between regional urbanization and the ecological environment. This p...

The study of networks and graphs carried out by topological measures performs a vital role in securing their hidden topologies. This strategy has been extremely used in biomedicine, cheminformatics and bioinformatics, where computations dependent on...

The reciprocal mean square (RMS) index of a graph G with edge set $E\left(G\right)$ is defined by $\mathcal{RMS}\left(G\right)={\sum }_{uv\in E\left(G\right)}{[{\left(d\left(u\right)\right)}^2+{\left(d\left(v\right)\right)}^2]}^{-1},$ where $d\left(u\right)$ and $d\left(v\right)$ are the degrees of vertices u and v, respectively. We first establish several bounds for the R...

There are several chemical indices that have been introduced in theoretical chemistry to measure the properties of molecular topology, such as distance-based topological indices, degree-based topological indices and counting-related topological indic...

Public transport accessibility (PTA) is an essential index for evaluating the efficiency of urban public transport resource and public service. Improving public transport accessibility is considered as the most effective way of alleviating urban cong...

The Randić index is among the most famous degree-based topological indices in chemical graph theory. It was introduced due to its application in modeling the properties of certain molecular structures and has been extensively studied. In this pa...

Healthcare accessibility (HCA) is directly related to the general well-being of citizens, and the HCA index model is widely used in HCA evaluation. However, the evaluation results of the HCA index model are rough and potentially misleading because it...

A reasonable definition of nodes load and capacity is essential for improving the robustness of scale-free networks against cascading failure, which has gained significant attention over recent years. This paper presents two methods for defining the...

The diminished Sombor index is a degree-based topological index that normalizes the Sombor contribution of each edge (defined as the Euclidean norm of the endpoint degrees) by the sum of those degrees, thereby making the index independent of graph si...

A topological index is a number that is connected to a chemical composition in order to correlate a substance’s chemical makeup with different physical characteristics, chemical reactivity, or biological activity. It is common to model drugs an...

In this report, we compute closed forms of M-polynomial, first and second Zagreb polynomials and forgotten polynomial for Jahangir graphs J_n,_m for all values of m and n. From the M-polynomial, we recover many degree-based topological indices such as...

This study explores the interaction mechanisms between population and environment systems within the context of high-quality development (HQD), providing empirical insights for developing countries navigating rapid urbanization. The existing literatu...

It is well-known that the necessary and sufficient condition for a connected graph to be unicyclic is that its omega invariant, a recently introduced graph invariant useful in combinatorial and topological calculations, is zero. This condition could...

Entropy is a thermodynamic function in chemistry that reflects the randomness and disorder of molecules in a particular system or process based on the number of alternative configurations accessible to them. Distance-based entropy is used to solve a...

Research on fusion modeling of high spatial and temporal resolution images typically uses MODIS products at 500 m and 250 m resolution with Landsat images at 30 m, but the effect on results of the date of reference images and the ‘mixed pixels’ natur...

Let G be a graph with set of vertices $V\left(G\right)$$\left(\right|V\left(G\right)|=n)$ and edge set $E\left(G\right)$. Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by $SO\left(G\right)$ and is defined as $SO=SO\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}\sqrt{d_G{\left(v_i\right)}^2+d_G{\left(v_j\right)}^2}$, where $d_G\left(v_i\right)$ is th...

For a graph $G=(V_G,E_G)$, a degree-based graphical index $G{I}_d$ takes the general form $G{I}_d={\displaystyle \sum _{xy\in E_G}}\varphi (d_x,d_y)$, where $\varphi$ is a symmetric map and $d_i$ is the degree of $i\in V_G$. For $\alpha \in \mathbb{R}$, if $\varphi ={\left(d_x{d}_y\right)}^{\alpha }$ (resp. $\varphi ={(d_x+d_y)}^{\mathrm{\&al}}$...

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In th...

The bismuth tri-iodide $(Bi{I}_3)$ is an inorganic compound. It is the result of the response of bismuth and iodine, which has inspired enthusiasm for subjective inorganic investigation. The topological indices are the numerical invariants of...

Recently, the exponential arithmetic–geometric index ($EAG$) was introduced. The exponential arithmetic–geometric index ($EAG$) of a graph G is defined as $EAG\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}{e}^{{\displaystyle \frac{d_i+d_j}{2\sqrt{d_i{d}_j}}}}$, where $d_i$ represents the degree of the vertex $v_i$...

Molecular fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (suc...

Due to significant advancements being made in the field of drug design, the use of topological descriptors remains the primary approach. When combined with QSPR models, descriptors illustrate a molecule’s chemical properties numerically. Number...

Topological indices are numeric quantities that describes the topology of molecular structure in mathematical chemistry. An important area of applied mathematics is the chemical reaction network theory. Real-world problems can be modeled using this t...

Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Then, the Sombor index of G is defined as $SO\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$. In this paper, we give some lemmas that c...

A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology a...

Graph entropy plays an essential role in interpreting the structural information and complexity measure of a network. Let G be a graph of order n. Suppose $d_G\left(v_i\right)$ is degree of the vertex $v_i$ for each $i=1,2,\dots ,n$. Now, the k-th degree-based graph e...

Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of configurations that molecules might take. Distance-based entropy is used to address a wi...

A fullerene is a cubic three-connected graph whose faces are entirely composed of pentagons and hexagons. Entropy applied to graphs is one of the significant approaches to measuring the complexity of relational structures. Recently, the research on c...

Fullerene is a cage-like carbon allotrope admitting a vast range of applications. Some of the important fullerenes are $C_{54},C_{58},C_{60},C_{70},C_{74},C_{76},C_{78},C_{80}{C}_{82},C_{84},C_{86},C_{90}$. The physical properties of fullerenes can be exhibited using the degree-based topol...