University of KashanIranian Journal of Mathematical Chemistry2228-648914320230901Introducing Two Transformations in Fullerene Graphs, Star and Semi-Star13514311397010.22052/ijmc.2023.252986.1722ENMeysam Taheri-DehkordiUniversity of Applied Science and Technology (UAST), Tehran, IRAN0000-0002-6085-2622Journal Article20230521A perfect star packing in a given graph $G$ can be defined as a spanning subgraph of $G$, wherein each component is isomorphic to the star graph $K_{1,3}$. A perfect star packing of a fullerene graph $G$ is of type $P0$ if all the centers of stars lie on hexagons of $G$. Many fullerene graphs arise from smaller fullerene graphs by applying some transformations. In this paper, we introduce two transformations for fullerene graphs that have the perfect star packing of type $P0$ and examine some characteristics of the graphs obtained from this transformation.https://ijmc.kashanu.ac.ir/article_113970_ac6a5fb04a4e203b1a9fd7e13c0f69e9.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648914320230901Deficiency Sum Energy of Some Graph Classes14516011397110.22052/ijmc.2023.252633.1698ENOmendra SinghDepartment of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India0000-0002-5224-9321Pravin GargDepartment of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India0000-0001-5836-6521Neha KansalDepartment of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India0000-0001-9577-1294Journal Article20230309In this paper, we introduce the concept of deficiency sum matrix $S_{df}(G)$ of a simple graph $G=(V,E)$ of order $n$. The deficiency $df(v)$ of a vertex $v \in V$ is the deviation between the degree of the vertex $v$<br /> and the maximum degree of the graph. The deficiency sum matrix $S_{df}(G)$ is a matrix of order $n$ whose $(i,j)$-th entry is $df(v_{i})+df(v_{j})$, if the vertices $v_{i}$ and $v_{j}$ are adjacent and $0$, otherwise. In addition, we introduce deficiency sum energy $ES_{df}(G)$ of a graph $G$ and establish some bounds for $ES_{df}(G)$. Further, deficiency sum energy of some classes of graphs are obtained. Moreover, we construct an algorithm and python(3.8) code to find out spectrum and deficiency sum energy of graph $G$.https://ijmc.kashanu.ac.ir/article_113971_6a0f0a71de5939bc3635ff1461755abd.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648914320230901Direct Mathematical Models for Estimating the Shelf Life of Second- and Zero-Order Degradation Relationships of Food and Drugs16116911397210.22052/ijmc.2023.252798.1715ENRabah AliKhalilDepartment of Chemistry, College of Science, University of Mosul, Mosul, Iraq0000-0002-5252-3335Journal Article20230424Recently, a direct model for evaluating the shelf life of lemon juice, depending on the loss of ascorbic acid concentration with time was derived by Al-Zubaidy and Khalil (Food Chem. 101 (2017) 254–259). By using this model, one could directly estimate the expiration date at any residual ratio and any specific temperature of the first-order degradation rate. But, in general, the kinetic model for degradation process is not limited to first-order reactions, as it could be second-, or zero-order according to the experimental data. Therefore, two direct models have been developed to evaluate shelf life based on these orders. These models have been verified using previously published results for vitamins C (zero-order) and D3 (2nd-order) in coconut powder. The results indicate that the prediction of shelf life for food, drugs and so on can also be evaluated directly for second-, and zero-order degradation processes using the developed models in the same way to that of pre-published for first order. A characteristic feature of the presented degradation orders is that the input data for determining the rate constant must be in percentage (\%) rather than the real concentration in contrast to that of first order. It was concluded that the use of the derived equations avoids the complications resulting from changing the unit of the rate constant with the change in the reaction order. In addition to the gained simplicity when dealing with the developed models.https://ijmc.kashanu.ac.ir/article_113972_e18171b090e6a0777799a7f2d2f6d065.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648914320230901On Extremal Values Of Total Structure Connectivity and Narumi-Katayama Indices on the Class of all Unicyclic and Bicyclic Graphs17118111397310.22052/ijmc.2023.252930.1716ENMohammad Ali ManianDepartment of Mathematics, Shahrekord Branch,
Islamic Azad University, Shahrekord, Iran0000-0002-3090-4313Shahram HeidarianDepartment of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran0000-0002-4515-0665Farhad Khaksar HaghaniDepartment of Mathematics, Shahrekord Branch,
Islamic Azad University, Shahrekord, Iran0000-0002-3510-8957Journal Article20230507The total structure connectivity and Narumi-Katayama indices of a simple graph $G$ are defined as $TS(G)={\prod_{{u}\in{V(G)}}}{\frac{1}{\sqrt {{d_{u}}}}}$ and $ NK(G)={\prod_{{u}\in{V(G)}}{{d_{u}}}}$ respectively, where $d _{u} $ represents the degree of vertex $ u $ in $ G $. In this paper, we determine the extremal values of total structure connectivity index on the class of unicyclic and bicyclic graphs and characterize the corresponding extremal graphs. In addition, we determine the bicyclic graphs extremal with respect to the Narumi-Katayama index.https://ijmc.kashanu.ac.ir/article_113973_c2cca9f1bb93fd555ece2099195149ad.pdfUniversity of KashanIranian Journal of Mathematical Chemistry2228-648914320230901Degree-Based Function Index of Graphs with Given Connectivity18319411397410.22052/ijmc.2023.252646.1699ENTomas VetrikDepartment of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa0000-0002-0387-7276Journal Article20230313We investigate the~ index $I_{f} (G) = \sum_{vw \in E(G)} f(d_G (v),d_G (w))$ of a graph $G$, where $f$ is a symmetric function of two variables satisfying certain conditions, $E(G)$ is the edge set of $G$, and $d_G (v)$ and $d_G (w)$ are the degrees of vertices $v$ and $w$ in $G$, respectively. Those conditions are satisfied by functions that can be used to define the general sum-connectivity index $\chi_{a}$, general Randi\'{c} index $R_{a}$, general reduced second Zagreb index $GRM_a$ for some $a \in \mathbb{R}$, general Sombor index $SO_{a,b}$, general augmented Zagreb index $AZI_{a,b}$ and by one other generalization $M_{a,b}$ for some $a, b \in \mathbb{R}$. The general augmented Zagreb index is a new index defined in this paper.<br /> <br />We obtain a sharp upper bound on $I_f$ for graphs with given order and connectivity, and a sharp lower bound on $I_f$ for $2$-connected graphs with given order. Our upper bound holds for $M_{a,b}$ and $SO_{a,b}$ where $a, b \ge 1$; $\chi_a$ and $R_a$ where $a \ge 1$; and $GRM_{a}$ where $a > -1$. Our lower bound holds for $M_{a,b}$ where $a \ge 0$ and $b \ge -a$; $SO_{a,b}$ where $a, b \ge 0$ or $a, b \le 0$; $AZI_{a,b}$ where $a \ge -2$ and $b \ge 0$; $\chi_a$ and $R_a$ where $a \ge 0$; and $GRM_{a}$ where $a > -2$.https://ijmc.kashanu.ac.ir/article_113974_bd3add0389466453d470906d1aaff5ad.pdf