University of KashanIranian Journal of Mathematical Chemistry2228-648917120260301Variational Formulation of Thermal Explosion Problem with Internal Heat Generation11111540710.22052/ijmc.2025.256937.2017ENIgorDonskoyMelentiev Energy Systems Institute SB RAS, 130 Lermontova st., Irkusk, Russian Federation, 664033Journal Article20250526The 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_115407_b3c20ca8f0a17e9f4bc4334da49725cb.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648917120260301Properties of Laplacian Eigenvalues of Some Bicyclic and Tricyclic Graphs133311541410.22052/ijmc.2025.256371.1984ENMohammadTariq RahimDepartment of Mathematics, Abbottabad University of Science \& Technology, Havelian, KPK, PakistanMuhammad AdnanAttaDepartment of Mathematics, Abbottabad University of Science \& Technology, Havelian, KPK, PakistanAfeefaMaryamDepartment of Mathematics, Abbottabad University of Science \& Technology, Havelian, KPK, PakistanFawadHussainDepartment of Mathematics, Abbottabad University of Science \& Technology, Havelian, KPK, PakistanJournal Article20250309The 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_115414_57aa30175c23a87b502d98724acbfb12.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648917120260301Harmonic-Arithmetic Index of Unicyclic Graphs with given Girth and Connected Graphs with Minimum Degree355111541510.22052/ijmc.2025.257071.2022ENJayavelPoojaDepartment of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of
Science and Technology, Kattankulathur, Tamil Nadu 603203, IndiaVenkatesanMuthukumaranDepartment of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of
Science and Technology, Kattankulathur, Tamil Nadu 603203, IndiaSureshElumalaiDepartment of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of
Science and Technology, Kattankulathur, Tamil Nadu 603203, IndiaSelvarajBalachandranDepartment of Mathematics, School of Arts, Sciences, Humanities, and Education, SASTRA Deemed University, Thanjavur 613401, IndiaJournal Article20250618Let 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_115415_39336345e5150c0ee6b45f519c13bbd5.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648917120260301On the Energy and Nullity of Non-Uniform Path and Cycle Semigraphs537511541910.22052/ijmc.2025.257303.2041ENSerin ElezabethJoyDepartment of Mathematics, Mar Athanasius College of Engineering (Autonomous), Kothamangalam, Ernakulam, 686666, Kerala, IndiaRajesh K. ThumbakaraDepartment of Mathematics, Mar Athanasius College (Autonomous), Kothamangalam, Ernakulam, 686666, Kerala, IndiaJournal Article20250727Graph 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_115419_a3ed3d892b4e24a2ba550a765a08cae5.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648917120260301Robust Numerical Approach for Solving Robin Boundary Value Problems779411543110.22052/ijmc.2025.256851.2008ENAhmed GamalAttaDepartment of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, EgyptSamehH. BashaDepartment of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt//Department of Mathematics, Faculty of Science, Galala University, Suez, 43511, Egypt//Scientific Research School of Egypt (SRSEG)Journal Article20250513In 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.https://ijmc.kashanu.ac.ir/article_115431_76a80efccde54a19152d9b933fcc377e.pdf