Graph descriptors such as normalized Laplacian energy, Sombor index, Mostar index and Zagreb index are often employed in various scientific contexts. The Sombor index is essential in chemical graph theory because it provides insights into molecule stability and reactivity by analyzing the distribution of atom degrees and distances. By measuring molecular graph topology, the Zagreb indices are vital tools in mathematical chemistry that help predict molecules’ physical and chemical characteristics. This manuscript aims to broaden the characterization of extremal graphs, achieving upper sharp bounds for the Mostar and first Zagreb index with given parameters such as cut edges and girth.
The first and second Zagreb indices are among the most studied vertex–degree–based graph invariants and play an important role in both chemical graph theory and extremal graph analysis. In this paper we establish new inequalities that directly relate these two indices under general structural assumptions.
Based on the Sombor index of graphs defined by Gutman in 2021, Shetty and Bhat defined the Sombor index of hypergraphs recently. Inspired by the work of Deng et al. [Int. J. Quantum Chem. 121 (2021) e26622] and Vuki´cevi´c [Math. Montisnigri. 50 (2021) 5-14], we characterize the extremal hypergraph with the upper and lower bounds of Sombor index for k-uniform chemical hypergraphs with n vertices and give the corresponding value of Sombor index. Furthermore, we characterize the extremal hypergraph with the upper bound of Sombor index for k-uniform chemical hypertrees with n vertices and give the corresponding value of Sombor index.
Total number of matchings is called Hosoya index in graph theory. Although Hosoya index was introduced in 1971, the relations between the Hosoya index of caterpillar graphs and Euler’s continuants were introduced in 2007. In this paper, we show that the Hosoya index of caterpillar graphs can be computed as product of 2 × 2 matrices. Moreover, we obtain that Hosoya index of the caterpillar graphs Z(Cn(x, . . . , x)) equals to (n + 1) − th Fibonacci polynomial and we show that the Hosoya index of the caterpillar graphs Z(Cn(x + 1, . . . , x + 1)) can be shown in a polynomial form with the coefficients as a Pell triangle of Reference number A038137 in OEIS. Finally, we use these relations in the computation of the Kekul´e number of a class Hn,x of benzenoid chains which have n segments of length x.
In this paper, we introduce a novel mathematical framework called Hybrid Soft-Neutrosophic Controlled Metric Space (HSNCMS), which integrates soft set theory, neutrosophic sets, and controlled metric spaces. We develop auxiliary lemmas and establish a Banach-type fixed point theorem with complete proof for HSNCMS and introduce the concept of T-controlled contraction with related results. Furthermore, as an application we prove the existence and uniqueness of equilibrium concentration profiles for a nonlinear chemical reaction network, where uncertainty arises from fluctuations in temperature, pressure, catalyst effects, and incomplete experimental data. The proposed model generalizes several existing structures and provides a flexible tool for handling uncertainty, indeterminacy, and parameterization simultaneously.
We show that there exist graphs with harmonic index less than their minimum maximal matching number. This disproves the TxGraffiti conjecture stating that the minimum maximal matching number of a connected graph is at most its harmonic index.
In this paper we mathematically investigate the kinetics underlying the hydrogenolysis of xylitol over a metal based catalyst by using ordinary differential equations. The primary motivation for mathematically investigating this reaction lies on the fact that some of the value added products from this reaction are sourced from fossil fuels, which are not environmentally friendly. They have been reported by different researchers that they are the primary drive of global warming which leads to climate change, which causes severe disturbances in the Earth´s natural systems and economies. Xylitol, on the other hand, which is readily available from biomass have been reported as an alternative raw material for some of the value added products. A reaction mechanism proposed from experimental data is used to formulate a system of ordinary differential equations which is then analyzed using some qualitative analysis tools from mathematics. Numerical simulations are performed to try and ascertain the long time behavior of the system’s solutions. Results showed that the solutions to the system approaches a lower dimensional invariant set with Xk = 0 for k = 1,2,...8 and Xk > 0 for k = 9,10,11. These numerical results were found to be in agreement with the experimental results from the chemistry point of view. Therefore, it is believed that this model can be extended for use in other sugar alcohols higher than xylitol, like sorbitol as means to maximize the yields of the desired products.
In 2013, Das and Mojallal published a new upper bound on graph energy. We now comment on the proof of this bound, and show that it is inferior to McClelland’s classic estimate.
Recently, Ali et al. [1] posed several open problems concerning extremal graphs with respect to the atom-bond sum connectivity in-dex. These problems involve characterizing graphs that attain the maximum ABS index within specific graph classes, including: (i) connected graphs with n vertices and p cut-vertices; (ii) connected graphs of order n with vertex k-partiteness vk(G) = r; and (iii) connected bipartite graphs of order n with a fixed vertex connectiv-ity κ. In this paper, we provide complete solutions to all of these problems.
The notions of vertex energy and centrality measures are significant graph invariants that describe the contribution of individual vertices to the total energy of graph. Unlike global energy measures, they provide vertex-level information and allow the identification of structurally significant vertices. In this article, we extend the notion of energy of a vertex (EG(v)) by defining the general vertex degreebased (VDB)vertex energy. Further, we study some particular types of this invariant, namely the first Zagreb vertex energy (M1EG(v)), second Zagreb vertex energy (M2EG(v)), forgotten vertex energy (FEG(v)), Sombor vertex energy (SOEG(v)) and atom-bond connectivity vertex energy (ABCEG(v)), derived from the corresponding VDB topological indices and their associated VDB index-weighted adjacency matrices. Furthermore, using the method proposed by Arizmendi et al., we compute these invariants for certain standard graphs. Subsequently, we perform a vertex-level regression analysis between the eigenvector centrality measure (Xi) and these particular VDB vertex energy invariants, with reference to each of the 18 octane isomers, through which we observe a strong correlation between these parameters, thereby establishing their significance.
In chemical systems, molecular properties such as electronegativity are often subject to uncertainty due to environmental fluctuations, measurement errors, or intrinsic variability. Fuzzy set theory provides a framework to model such uncertainties, allowing chemical descriptors to be expressed as degrees of membership rather than fixed values. This paper develops a rigorous order-theoretic framework for studying fuzzy structures arising from property-induced partially ordered sets (posets) in molecular systems. By extending classical poset theory through fuzzy relations, we formalize graded comparability and inclusion principles that naturally occur in chemical and molecular structures. The proposed approach preserves essential order-theoretic properties while allowing flexible representation of uncertainty.
A basic representation of any real molecule is a finite cloud of unordered atoms, many of which are chemically indistinguishable. A natural equivalence on point clouds in any metric space is defined by isometries that are distance-preserving transformations. In a Euclidean space, any isometry is a composition of translations, rotations, and reflections. If points are ordered, the isometry class of this cloud is uniquely determined by the matrix of all pairwise distances. If m points are unordered, a naive metric based on distance matrices needs exponentially many m! permutations. We define a complete invariant for n-dimensional clouds of m unordered points under rigid motion, which distinguishes all mirror images in Rn. The key challenge was to design a distance on invariant values that is Lipschitz continuous under noise and computable in a polynomial time of cloud sizes, for a fixed dimension n.
The geometric quadratic (GQ) index is a recently introduced degree-based topological descriptor, and Kumar et al. observed that it is potentially a very good molecular descriptor. In this paper, we characterize the extremal graphs (chemical) and trees concerning the geometric quadratic index of a given order and size. Then, we determine the -vertex trees, unicyclic and bicyclic graphs with the maximum, the second, the third, the fourth, the fifth, and the sixth maximum geometric quadratic indices.
