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.
