In this work, we introduce and develop spectral collocation techniques for solving second-order differential equations (SODEs) arising in chemical processes such as catalytic reactions, diffusion-reaction systems, and thermal conduction in reactive media, where Robin boundary conditions naturally emerge due to combined flux and concentration constraints. The proposed approach can be roughly represented as a truncated series of modified shifted fourth-kind Chebyshev polynomials (4KCPs). The unknown expansion coefficients are determined using the spectral collocation method. Collocation nodes were the shifted 4KCPs roots. The resulting nonlinear algebraic system is solved efficiently using Newton's method. We present a theorem that shows the truncation error rapidly converges with respect to the number of retained modes. The method's applicability and effectiveness are demonstrated using some numerical examples. (c) 2026 University of Kashan Press. All rights reserved
This manuscript examines two categories of fractional optimal control problems (FOCPs) with fractional system and delay fractional system constraints. This scheme is based on the general Lagrange scaling functions (GLSFs), which can generate both orthogonal and non-orthogonal scaling functions by selecting different Lagrange nodes. Notably, the method is designed to be applied without initially choosing specific Lagrange nodes; instead, we leverage the potential advantages of GLSFs to develop new methods by considering various Lagrange nodes. Additionally, a general Riemann-Liouville fractional integration operational matrix (GR-LOP) and a general delay operational matrix (GDOP) are proposed for the considered functions. Next, by combining these operational matrices and the Gauss-Legendre integration method, we transform the original problems into systems of algebraic equations. To demonstrate the effectiveness of the proposed GLSF method, five numerical examples are provided. (C) 2026 University of Kashan Press. All rights reserved.
The Laplacian energy (LE) and the Laplacian energy-like (LEL) have recently been proposed based on molecular graph analogues of the total pi-electron energy E. Both energies have been widely studied recently because of their wide range of applications. In the present work, exact expressions of the Laplacian energy and the Laplacian-like invariants of bicyclic and tricyclic molecular graphs in terms of their orders have been obtained. We also compute these expressions for the complements of these classes of graphs. It is shown that LEL is strictly less than LE for these classes of molecular graphs, but for their complements the inequality is the opposite.
Let G be the finite, simple, and connected graph with a vertex set as V (G) and an edge set as E(G). The harmonic-arithmetic index of graph G is defined as HA(G) = Sigma(P rho phi is an element of E(G)) 4d(rho)d(phi)/(d(rho)+d(phi))(2) where d(rho) denotes the degree of the vertex rho and rho phi denotes the edge. Let U-eta ,U-g be the set of unicyclic graphs with eta vertices and given girth g. Let G(eta ,delta )be the set of simple connected graphs with eta vertices with minimum degree S. In this article, we present the maximum and second-maximum harmonic-arithmetic index of unicyclic graphs with a given girth and determine their corresponding graphs. The obtained results remain valid when the analysis is confined to the class of chemical unicyclic graphs. Further, we obtain extremal graphs in G(eta ,delta) for which the HA index reaches its smallest value, or we provide a lower bound, for delta >= [delta(0)], with delta(0) = p(0)(eta - 1), where p(0 )approximate to 0.23606 is the distinct positive root of the expression p(2) + 4p-1 = 0. We demonstrate that the extremal graphs are regular graphs of degree delta when delta or eta is even.
Graph energy, originating in H & uuml;ckel molecular orbital theory, remains central to mathematical chemistry. Motivated by heterogeneous linear and cyclic molecular structures, we study non-uniform path and cycle semigraphs, where original edges are subdivided by n(i)>= 1 middle vertices. We show the adjacency matrix decomposes into a symmetric tridiagonal core, whose spectrum comprises all non-zero eigenvalues, plus zero rows from middle vertices. For paths, a continuant recurrence for the characteristic polynomial and parity arguments yield spectral symmetry and precise nullity conditions. For cycles, a wraparound determinant formula characterizes when the spectrum is symmetric about zero and provides exact criteria for the presence and multiplicity of specific zero eigenvalues. Consequently, the energy of each semigraph equals the energy of its core matrix, yielding clean expressions for energy and nullity from the {n(i)} parameters. Uniform cases arise as immediate corollaries and are consistent with spectral invariants in chemically inspired models.
The article considers the problem of thermal stability in plane symmetry with an exothermic chemical reaction and constant heat release. The dependence of the critical reactivity on the intensity of heat release is investigated. Differential and variational formulations are considered; for the latter, an approximate analytical solution is given that relates the parameters of the problem for critical conditions. A simple Rayleigh-Ritz procedure results in a set of equations expressing the temperature distribution in terms of polynomials. The ignition boundary can be found through second derivatives of the integral, which can be evaluated using some simplifications that are typical for combustion theory. The results are reduced to simple approximations that can be used to estimate the ignition limits in systems with combined heat release. (c) 2026 University of Kashan Press. All rights reserved.
A fullerene graph is a 3-connected, planar cubic graph where each face is either pentagonal or hexagonal. This paper investigates the symmetric division eccentric index (SDE) of fullerenes, particularly focusing on two infinite classes F-10n and F-12n. We establish several bounds for fullerene graphs. We also explore the automorphism group actions on the vertices and edges of these fullerene graphs, establishing a relation between edge orbits and their eccentricities. General formulas for calculating SDE-indices are derived for F-10n, when n >= 8 and F-12n, when n >= 10. Furthermore, we present a new approach to compute the SDE-index and then implement the method to obtain general formulas for the SDE-index of the given classes of fullerenes. (c) 2025 University of Kashan Press. All rights reserved.
This paper presents a numerical method for solving a class of nonlinear multi-order fractional differential equations using the first-kind Chebyshev polynomials. The proposed approach is based on a collocation framework that incorporates operational matrices of derivatives specifically tailored to the spectral properties of the Chebyshev polynomials on the interval [0, 1]. Two cases of interest are considered: the classical case with nu = 2 and lambda = 1, and the fractional-order case with 1 < nu <= 2 and 0 < lambda <= 1. To ensure high accuracy, an appropriate set of the shifted Chebyshev basis functions that satisfy the boundary conditions is utilized. The Caputo definition of fractional derivatives is adopted to handle the fractional operators. The resulting nonlinear algebraic system is solved efficiently using Newton's method. Numerical experiments confirm the proposed method's efficiency, stability, and accuracy in comparison with existing techniques.
A family of complex amplified Stormer methods is studied for solving initial value problems of second-order differential equations with periodic or orbital solutions. The new complex amplified Stormer methods depend upon a parameter w > 0, vanish its complex amplifier, and integrate precisely algebraic polynomials. We believe that each method category (Stormer method is one of them) has its complex amplifier. When finding the coefficients of the Stormer methods, if the imaginary and real parts of the complex amplifier, if necessary, their derivatives are equal to zero, high-capability methods are obtained. The principal local truncation errors of the new explicit Stormer methods are addressed. Their stability regions are depicted in a plane where the vertical axis is the problem frequency and the Horizontal axis is the method frequency. A collection of numerical examples illustrates the success of the new family of complex amplified Stormer methods in addressing the Schrodinger equation and other related problems. The advantage of the new methods is showcased by discussing their relevance to some issues in chemistry. (c) 2025 University of Kashan Press. All rights reserved.
In the kinetic theory of chemical processes, cyclic reactions play a significant role. This study addresses a four-component non-linear Brusselator reaction-diffusion model for cyclic reactions to investigate the existence of periodic wave solutions to the model. The local behavior of the non-diffusive model solutions is investigated first. Then, the existence and uniqueness of the solution in the diffusive model are discussed. Furthermore, the emergence of periodic wave solutions in the diffusive model is also analyzed analytically. To validate our theoretical investigation, we also perform some numerical simulations in one and two space dimensions of the diffusive model. To demonstrate the originality of this study, we conclude by summarizing our findings and drawing comparisons with previous research. (c) 2025 University of Kashan Press. All rights reserved.
Our main interest in this paper is the study of the Hosoya index Z (G; phi) of weighted graphs (G; phi), when G is a hexagonal chain with weight function induced by a vertex-degree-based topological index phi. Recall that a hexagonal chain is a special type of hexagonal systems, natural graph representations of benzenoid hydrocarbons. On the other hand, vertex-degree based topological indices are (molecular) graph descriptors which play a significant role in chemical graph theory. Concretely, if G is a hexagonal chain and phi is a vertex-degree-based topological index, we give a method to compute Z (G; phi) in terms of products of namely four types of 4 x 4 matrices associated to phi. As a consequence, under certain conditions on phi, we show that the phi-weighted linear hexagonal chain attains the minimal value of the Hosoya index, among all phi weighted hexagonal chains. (c) 2025 University of Kashan Press. All rights reserved.
A polynomial function that provides information about the molecular structure of a graph is known as the M-polynomial of a graph. This polynomial helps us to understand the characteristics of chemical compounds and their relationships. Very recently, in 2024, the elliptic Sombor (ESO), reduced elliptic (RE) and modified reduced elliptic (mRE) indices of a graph were proposed and their values were calculated for some standard graphs, jagged-rectangle benzenoid systems and polycyclic aromatic hydrocarbons. In this work, we establish closed derivation formulas for the above-mentioned elliptic-type indices of a graph based on its M-polynomial. Moreover, we enumerate the elliptic-type indices of the above family of chemical graphs. (c) 2025 University of Kashan Press. All rights reserved.
Let G be a simple graph with vertex set V (G) = {v(1),v(2), ... , v(n)}. The Randi & cacute; matrix of G, represented as R(G), is defined as the n x n matrix whose (i, j)-entry is (d(i)d(j))(-1/2 ) if v(i )and v(j )are adjacent and 0 otherwise. The Randi & cacute; energy of graph G is the sum of absolute values of the eigenvalues of R(G). In this study, we determine the Kragujevac trees with a fixed degree and fixed order that have maximal and minimal Randi & cacute; energy. Additionally, we obtain upper and lower bounds for the Randi & cacute; energy of these trees. (c) 2025 University of Kashan Press. All rights reserved.
This study investigates the relationship between the structural properties of 22 analgesic molecules, including opioids and non-steroidal anti-inflammatory drugs (NSAIDs), and their physico-chemical properties and side effects. Opioids are effective for managing moderate to severe pain but carry risks of addiction and dependence, while NSAIDs are widely used for their analgesic, antipyretic, and anti-inflammatory properties, with risks primarily linked to gastrointestinal and cardiovascular complications. Using graph theoretical approaches, we modeled these molecules as molecular graphs, calculating ten distance-and degree-based topological indices. The analysis aims to identify structural patterns that correlate with the drugs' therapeutic effects and adverse reactions. By leveraging computational tools such as Sage, we explore how molecular topology can influence the pharmacological properties of analgesics, offering insights into the design of new analgesics with optimized efficacy and safety profiles. The results provide insight into connections between molecular structure and clinical outcomes, contributing to more effective and safer approaches to pain management.
Herein, we introduce a novel computational approach to find a solution of variable-order fractional differential equations (VO-FDE). The desired method is proposed by combining fractional-order Pell functions, the Ritz and collocation methods. First, we define a new set of hybrid functions named fractional-order Pell hybrid functions. Next, to approximate the solution of VO-FDEs, we obtain an extra pseudo-operational matrix of the Caputo variable-order derivative. Then, by using the Ritz method, the operational matrix approach, and the collocation method, the problem is transformed to a system of algebraic equations, which is solved by Newton's iterative method. The error estimation of the proposed method is also examined. Finally, some examples (especially in chemistry) are presented to illustrate the effectiveness of this method. (c) 2025 University of Kashan Press. All rights reserved.
A (k, 6)-fullerene graph refers to a planar 3-connected cubic graph whose faces are k-gons and hexagons. The current study involves calculating the local metric dimension for specific (k, 6)-fullerene graphs, where k takes values in the set {3, 4, 5}.
Degree distance D'(G) is an important molecular descriptor which provides valuable insights into the connectivity and properties of molecular graphs, making it a powerful tool in diverse areas of chemical graph theory. This descriptor has attained much attention in the recent past for its broad range of applicability in different problems of chemical graph theory. Ordering of graphs with certain parameters allows chemists to identify patterns and trends of different chemical compounds and as a result predict their reaction behavior accordingly. In this paper, the first sixteen tricyclic graphs are presented which have minimum degree distances (in ascending order) if n >= 31. (c) 2025 University of Kashan Press. All rights reserved.
The notion of hyperstructures is a generalization of algebraic structures. This notion was first introduced by Marty in 1934. Hyperstructures have many applications, such as in biology, physics, cryptography, and chemistry. This paper focuses on the application of hyperstructures in chemistry, especially in chemical reactions. In 2022, Al-Tahan and Davvaz finalized the results of chemical hyperstructures for chemical elements that have four oxidation states. Motivated by this research, this paper aims to investigate algebraic hyperstructures in some elements that have five oxidation states, that is, neptunium, rubidium, and plutonium. Furthermore, the chemical interpretation of these chemical elements also is provided in this paper. (c) 2025 University of Kashan Press. All rights reserved.
The first general multiplicative Zagreb index P1a(G) is the product of the degree of each vertex v in G, raised to the power a and the second general multiplicative Zagreb index P2a(G) is the product of the degree of each vertex v in G, raised to the power a times the degree of v, where a is a non-zero real number. In this study, we present bounds on the general multiplicative Zagreb indices for trees and unicyclic graphs. We also provide bounds for the first general multiplicative Zagreb index for trees. Additionally, we identify all the extremal graphs for each bound mentioned as best as possible. (c) 2025 University of Kashan Press. All rights reserved.
In this article, we discuss the additively weighted Mostar index, an innovative topological measure that extends the traditional Mostar index by incorporating edge weights computed as the sum of the degrees of their end point vertices. We focus on its application within the set Un comprising all unicyclic graphs of order n. Our study rigorously establishes the first two sharp lower and upper bounds for this index across graphs in Un. Additionally, we analyze the additively weighted Mostar index of Cartesian product graphs and investigate its properties across various graph classes. Furthermore, we demonstrate the practical utility of this index by comparing its effectiveness against eight other distance-based topological indices in predicting chemical properties of octane isomers. index outperforms or matches the predictive capabilities of other indices in linear models with these chemical properties, highlighting its potential in quantitative structure-property relationships. This research significantly contributes to both graph theory and chemical informatics by showcasing the unique advantages of the additively weighted Mostar index in structural analysis and predictive modeling.
