https://ijmc.kashanu.ac.ir/ Iranian Journal of Mathematical Chemistry en daily 1 Sun, 01 Mar 2026 00:00:00 +0330 Sun, 01 Mar 2026 00:00:00 +0330 https://ijmc.kashanu.ac.ir/article_115407.html ‎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. https://ijmc.kashanu.ac.ir/article_115414.html ‎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. https://ijmc.kashanu.ac.ir/article_115415.html ‎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 $HA(G) = \sum\limits_{\rho\phi \in E(G)} {\dfrac{{4{d_\rho}{d_\phi}}}{{{{({d_\rho}‎ + ‎{d_\phi})}^2}}}}$ where $d_\rho$ denotes the degree of the vertex $\rho$ and $\rho\phi$ denotes the edge‎. ‎Let $U_{\eta,\mathfrak{g}}$ 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 $\delta$‎. ‎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 $\delta \geq\left\lceil \delta_0 \right\rceil$‎, ‎with $\delta_0 = p_0(\eta-1)$‎, ‎where $p_0 \approx 0.23606$ 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. https://ijmc.kashanu.ac.ir/article_115419.html ‎Graph energy‎, ‎originating in H\"uckel 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 $n_i \ge 1$ 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. https://ijmc.kashanu.ac.ir/article_115431.html ‎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.