Topological indices are numerical descriptors derived from the structure of a molecular graph and provide valuable information about the chemical, physical, and biological properties of molecules. The diminished Sombor index (DSO) of a graph G is defined as DSO(G)=∑ _v_i v_j ∈ E(G)√(d( v_i) ^2+d( v_j) ^2)/d( v_i) +d( v_j) , where d(v_i) denotes the degree of the vertex v_i and E(G) is the edge set of G. The characterization of extremal chemical trees with respect to topological indices is a well-established problem in chemical graph theory. Cruz et al. [1] provided a set of sufficient conditions for such characterizations. While most degree-based indices satisfy these conditions, the DSO index does not, making the problem of identifying extremal chemical trees particularly challenging. In this work, we completely resolve this problem and further investigate the applicability of DSO in structure–property modeling using chemical tree datasets. The results demonstrate that DSO exhibits strong predictive performance for enthalpy of vaporization, outperforming several well-known degree-based indices in both direct evaluation and 5-fold cross-validation analyses. These findings strengthen the claims made by Movahedi et al. [2].

The progression of 2D novel, metal trihalides MX3, has been learned Dirac halfmetallicity topological spintronic possessions because of their interest that are inherited magnetic processes reflected by 2D metal trihalide. Recently a new approach has been introduced to calculate the degree-based topological indices of chemical compounds, called the M-polynomial. It gives nice and good results of the topological indices of the corresponding chemical compounds. Then one can use these results to correlate the chemical compounds with their chemical properties and bioactivities. The main interest in this article is the computationsand comparisons of the general form of M-polynomial of two types of Subjclass[2010] 05C09, 05C92 metal trihalides.Then,the general form of obtained M-polynomials is used to determine the K d m tal t ih lid t p l gi l i di M p l l degree-based topological indices of these metal trihalides. In the end, graphical comparisons m i 2D plot of the M-polynomial and the topological indices of the metal trihalides are described. Mainly, a comparison among the degree-based indices of metal trihalides is performed with the help of 2D-chart plots and numerical tables.

One of the primary objectives in spectral graph theory is to explore how a graph’s structural properties are reflected in the algebraic features of its associated matrices. Among the fundamental tools in this field are the adjacency matrix and graph energy, both of which hold significant theoretical and practical importance. A topological index is a numerical descriptor derived from a molecular graph that captures information about its structure and connectivity, and is commonly used to correlate molecular structure with various physical, chemical, or biological properties. It is considered as numerical molecular descriptor. Degree-based molecular descriptors, in particular, have been extensively studied in the literature. For each such descriptor, a corresponding modified adjacency matrix and an associated graph energy can be defined. This study primarily investigates these extended graph energies based on several well-known degree-based descriptors, emphasizing their usefulness in modeling structure–property relationships through regression analysis.

The spectral graph theory investigates the relationships between combinatorial qualities of graphs and algebraic properties of related matrices. The adjacency matrix is currently undergoing significant modification as a result of its well-developed theoretical and application standpoint. The present work deals with one such extension of the adjacency matrix. We propose here the neighborhood Sombor matrix corresponding to the well-known Sombor index. We compute the neighborhood Sombor spectrum of some benchmark graphs. Lower and upper bounds of the spectral radius (zeta 1) are derived with identifying extremal graphs. Moreover, extremal trees are characterized in view of spectral radius, where path and star graphs yield minimal and maximal structures, respectively. The role of zeta 1 in structure-property modelling is also demonstrated. The isomer-discrimination ability of zeta 1 is found to be better than that of some well-known descriptors.

The Mostar invariants are newly introduced bond-additive, distance-related descriptors that compute the degree of peripherality of specific edges as well as the entire graph. These invariants have attracted significant attention in both classical applications of chemical graph theory and studies of complex networks. They have proven to be useful for exploring the topological aspects of these networks. For a graph ℋ, the edge Mostar index Moe is defined as the sum of the magnitudes of the differences between mℋ(x) and mℋ(g) across all edges xg of ℋ. Here, mℋ(g) (or mℋ(x)) represents the cardinality of the edges in ℋ that are closer to g (or x) than x (or g). In this paper, we determine the trees that maximize and minimize the edge Mostar index for fixed order, diameter, and number of pendent vertices. Sharp upper and lower bounds for this index are established, and the corresponding extremal trees are characterized. Moreover, the correlation of the edge Mostar index with certain physicochemical properties is examined.

A conjugated tree is defined as a tree that possesses a perfect matching. A chemical tree is a tree in which the degree of every vertex is at most 4. Let T be a conjugated chemical tree with edge set E. For any vertex u in T, let d(u) denote its degree. This study investigates a family of graph invariants defined as B (T) pound = & sum; (d(u) pound, d(v)), where uv is an element of E pound is a symmetric, real-valued function of the degrees of adjacent vertices of T. The principal objective is to determine those trees, among all conjugated chemical trees with a fixed order, that minimize or maximize B pound, subject to specific assumptions on the function pound.

One of the well-studied topics in extremal graph theory is the identification of extremal unicyclic graphs with respect to topological indices. Cruz, Rada, and Sanchez [MATCH Commun. Math. Comput. Chem. 88 (2022) 481-503] proposed a unified framework for identifying extremal unicyclic graphs with respect to degree-based topological indices, given the graph order. The exponential arithmetic-geometric index (EAG) is a degree-based index, which is defined for a graph G as EAG(G) = Sigma vivj is an element of E(G) dG(vi)+dG(vj)/e2 root dG(vt)dG(vj), where dG(vi) represents the degree of the vertex vi, and E(G) denotes the graph's edge set. The EAG index was excluded from the aforementioned framework due to its unique counting characteristics. Consequently, determining maximal unicyclic graphs for EAG was identified as an open problem in the same study. This work aims to address this gap by providing a comprehensive solution. We identify the maximal unicyclic graph for EAG with given graph order n. In addition, we explore sharp upper and lower bounds of EAG for trees as functions of graph order. The extremal chemical trees are also generated. Furthermore, we report regression relationships between EAG and the physicochemical properties of octanes.

Schizophrenia is a long-term, serious mental health condition that affects how a person thinks, perceives, and behaves. This disorder often results in substantial difficulties in social interactions and work performance. Individuals with schizophrenia might appear disconnected from reality, causing significant distress both for themselves and their Friends. Although symptoms of schizophrenia can vary from person to person, they typically fall into three main categories: cognitive, negative, and psychotic. Creating computational tools to find and develop drugs for schizophrenia has more interest in the past few years. Regardless of the significant developments in drug design, the fundamental approach still uses topological descriptors. Topological indices are used to estimate the bioactivity of chemical compounds in QSAR/QSPR studies. In general, with the use of the quantitative structure-property relationship (QSPR), topological indices are numerical values that are connected to the chemical drug structures and are used to predict their reactivity, stability, and properties. This work focuses on calculating different eccentric indices (EIs), developing a regression model for thirteen anti-schizophrenia drugs, and applying statistical methods to establish a linear regression relationship between QSPR correlating properties and eccentric indices. Statistical analysis shows that p-values less than equals 0.05, f-test value (>2.5), and values of correlation r are greater than 0.7 validate the calculations. The correlation coefficient (r2) is a convenient tool for evaluating the QSPR models' quality. r2>0.7 is essential for a good QSPR model. The p-values show the significance of the results, while the correlation coefficient values show the accuracy of the results. In order to fit regression models for the calculated eccentric index values, eight physicochemical properties of anti-schizophrenia drugs are examined. Drug properties like molar refractivity (cm3), refractive index (cm3), enthalpy (kJ/mol), melting, boiling and flash points (°C), complexity, and molecular weight are all more effectively estimated by the QSPR model. By examining actual and estimated values for the drugs, the results are verified.

The study of topological descriptors is essential for understanding the underlying structures of graphs and networks. Numerous numerical descriptors associated with graphs have been used to analyze their overall structure. In this analysis, degree-based topological descriptors hold a significant place. The Euler Sombor index of a graph Gamma is a significant vertex degree-based topological index related to the Sombor index, showing a strong correlation with the physicochemical properties of octanes. It represents the perimeter of an ellipse, with focal points recognized as the degree-point and the dual-point of a pair of connected vertices in Gamma. Nowadays, finding extremal results with respect to various graph indices for fixed graph parameters has become an important and engaging area of research in extremal graph theory. In this article, we explore the first and second maximum Euler Sombor index of a tree with a fixed diameter d >= 4. The extremal trees are also identified. Furthermore, we provide the ordering of trees for d=3. In addition, we identify the maximal unicyclic graph when the graph order and diameter are given.

This commentary addresses flaws discovered in Munir et al.’s recent paper [VI] on the M-Polynomial and associated topological descriptors of dendrimers. Our examination reveals inconsistencies in numerous critical aspects of their work. First, the figures representing the nanostar dendrimer formations must be more accurate. This means that the pictures used to depict the production of these molecules need to be revised. Second, flaws exist within the formulae used for the computations. Finally, based on incorrect formulae and figures, the claimed computational findings must be corrected. We suggest a complete correction approach to ensure the correctness of Munir et al.’s [VI] findings on nanostar dendrimers. It includes supplying corrected figures that accurately portray the nanostar dendrimer structures and removing the inconsistencies in the previous images. In addition, we will use revised notations to ensure that all parameters used in computations are explicitly and consistently described to eliminate ambiguities. Furthermore, we will provide the Accurate Formula for the Symmetric Division Index, which is critical for achieving precise results. Finally, all topological indices will be recalculated using these modifications with the revised figures and formulae to represent the genuine values. These corrections are necessary to give the validity of the results and pave the path for future research in this field.

The complementary second Zagreb index of a graph G is defined as cM(2)(G)=Sigma(uv is an element of E(G))|(du(G))(2)-(d(v)(G))(2)|, where d(u)(G) denotes the degree of a vertex u in G and E(G) represents the edge set of G. Let G* be a graph having the maximum value of cM(2) among all connected graphs of order n. Furtula and Oz [MATCH Commun. Math. Comput. Chem. 93 (2025) 247--263] conjectured that G* is the join K-k+(K) over bar (n-k) of the complete graph Kk of order k and the complement (K) over bar (n-k) of the complete graph (K) over bar (n-k) such that the inequality k [n/2] holds. We prove that (i) the maximum degree of G* is n-1 and (ii) no two vertices of minimum degree in G* are adjacent; both of these results support the aforementioned conjecture. We also prove that the number of vertices of maximum degree in G*, say k, is at most -2/3n+3/2+1/6 root 52n(2)-132n+81, which implies that k< 5352n/10000. Furthermore, we establish results that support the conjecture under consideration for certain bidegreed and tridegreed graphs. In the aforesaid paper, it was also mentioned that determining the k as a function of the n is far from being an easy task; we obtain the values of k for 5 <= n <= 149 in the case of certain bidegreed graphs by using computer software and found that the resulting sequence of the values of k does not exist in "The On-Line Encyclopedia of Integer Sequences" (an online database of integer sequences).

The Harary index and the Wiener index are two well-studied topological indices in chemical graph theory. Quite recently, the graphs attaining the minimum value of the product of the Harary and Wiener indices were characterized in [E. Azjargal, B. Horoldagva, I. Gutman, Minimum of product of Wiener and Harary indices, MATCH Commun. Math. Comput. Chem. 92 (2024) 65-71] over the class of all connected graphs of a fixed order and size. The present paper provides a generalization, involving Wiener-type topological indices and their reciprocals, of the aforementioned result.

The general $ Z $-type index is a molecular descriptor, introduced recently by Chen and Lin [Discrete Optim., 50 (2023), 100808], which generalizes several well-known molecular descriptors, including the (general) sum-connectivity index and (general) Platt index. The primary objective of the current paper is to study the largest value of the general $ Z $-type index of graphs in the class of all fixed-order trees (and chemical trees) with a particular number of segments.

In graph theory and mathematical chemistry, a topological index is a numerical molecular descriptor that quantifies the structural characteristics of a molecule without considering its three-dimensional structure. The structure-property relationship of chemical compounds can be numerically revealed using topological indices, eliminating the need for wet laboratory testing. Given its strong correlation with molecular properties and activities, the inverse symmetric division degree index (ISDD) is a well-established measure. In this work, we begin with mathematical exploration of its exponential variant (EISDD). The lower and upper bounds of EISDD for various families of graphs, including general connected graphs, trees, and bipartite graphs, are estimated. Corresponding extremal graphs are also examined. Finally, the role of the EISDD index in structure-property relationship modelling is analysed. The EISDD index is found to model different properties of octanes and some medicinal chemicals with significant accuracy.

. The basic idea of this paper is to address flaws discovered in Naz et al.'s recent paper [4]. In our examination, we discussed many incorrect aspects in the findings of main results. First, we will use revised notations to ensure that all parameters used in computations are consistently described to eliminate ambiguities. In addition, we give the corrections and correct approach to the expected results of multiplicative K Banhatti and multiplicative hyper K Banhatti indices of random polyphenyl chain and spiro chain. And the average value of the multiplicative K Banhatti and multiplicative hyper K Banhatti indices for the set of random poly-phenylene and spiro chains have been corrected. Moreover, we will provide the accuracy of the comparison between the expected results of multiplicative K Banhatti and multiplicative hyper K Banhatti indices for polyphenyl and spiro chains. We suggest a complete correction approach to ensure the correctness of Naz et al.'s [4] findings on the expected values of multiplicative K Banhatti indices for random poly-phenyl chain and spiro chain. These corrections are critical to establishing the validity of the findings and paving the path for future advances in this discipline.

A model for mobility of robots keeping the property of uniquely recognizing the vertices of a given network is considered in this work. This is made in order to detect failures or intruders, by means of dynamic vectors of distances to the set of mobile robots. We consider the smallest set of robots that can be placed in a set of nodes of a network that forms a resolving set, which is a structure of a graph such that it uniquely recognizes all the vertices of the graph by using distances. We are then focused on allowing such robots to move from one vertex to another adjacent one, through the edges of the whole graph. At each performed movement we require that the new set of covered nodes forms a resolving set. This process allows the robots to recognize all the vertices of the graph, independently on the position in which they are located. In this sense, the notion of mobile metric dimension is introduced in this article, and the study of its primary combinatorial properties is initiated. We relate this parameter with the classical metric dimension and the resolving number of graphs and compute its value for several graph classes.

Chemical graph theory plays a crucial role in mathematical chemistry by using graph invariants to represent chemical phenomena in a mathematical framework. Topological descriptors, which are graph invariants formed from molecular graph representations of chemical compounds, are employed in QSPR and QSAR studies. The Zagreb connection indices are topological descriptors which are used to analyze graphs based on connection cardinality. They were introduced in 1972 to calculate the total electron energy of alternate hydrocarbons. Recently, they have gained renewed attention due to their ability to provide a more precise correlation for the physicochemical characteristics of different compounds compared to basic Zagreb indices. In this article, we aim to analyze and establish the correlations of the first Zagreb connection index and the total domination number of trees. Our focus is on providing a detailed description of trees having the largest first Zagreb connection index among trees that have a specific total domination number. (c) 2025 Published by Elsevier B.V.

Consider a graph $G$ and a real-valued function $f$ defined on the degree set of $G$. The sum of the outputs $f(d_v)$ over all vertices $v\in V(G)$ of $G$ is usually known as the vertex-degree-function indices and is denoted by $H_f(G)$, where $d_v$ represents the degree of a vertex $v$ of $G$. This paper gives sharp bounds on the index $H_f(G)$ in terms of order and size of $G$ when $G$ is connected and has the maximum degree at most $4$. All the graphs achieving the derived bounds are also determined. Bounds involving several existing indices - including the general zeroth-order Randi\'c index and coindex, the general multiplicative first/second Zagreb index, the variable sum lodeg index, and the variable sum exdeg index - are deduced as the special cases of the obtained ones.

The concept of the weighted Mostar invariant is a mathematical tool used in chemical graph theory to study the stability of chemical compounds. Several recent studies have explored the weighted Mostar invariant of various chemical structures, including hydrocarbons, alcohols, and other organic compounds. One of the key advantages of the weighted Mostar invariant is that it can be easily computed for large and complex chemical structures, making it a valuable tool for studying the stability of a wide range of chemical compounds. This notion has been utilized to build novel approaches for forecasting chemical compound stability, such as machine learning algorithms. The focus of the paper is to demonstrate the weighted Mostar indices of three specific nanostructures: silicon dioxide (SIO2, poly-methyl methacrylate network (PMMA(s)), and melem chains (MC(h)). The authors seek to provide the findings of their investigation of these nanostructures using the weighted Mostar invariant.

Covalent organic frameworks are a novel class of porous polymers, notable for their crystalline structure, intricate frameworks, defined pore sizes, and capacity for structural design, synthetic control, and functional customization. This paper provides a comprehensive analysis of graph entropies and hybrid topological descriptors, derived from geometric, harmonic, and Zagreb indices. These descriptors are applied to study two variations of Marta covalent organic frameworks based on contorted hexabenzocoronenes. We also conduct a comparative analysis using scaled entropies, offering refined tools for assessing the intrinsic topologies of these networks. Additionally, these hybrid descriptors are used to develop statistical models for predicting graph energy in higher-dimensional Marta-COFs.