Iranian Journal of Mathematical Chemistry
Volume:14 Issue: 3, Summer 2023

‎A 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.

In 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 $v$‎‎ 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 $G$‎.

‎Recently‎, ‎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.

‎The 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.

‎We 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. ‎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 $a >‎ -‎2$.