In the work, author’s presents a very significant and important issues related to the health of mankind’s. Which is extremely important to realize the complex dynamic of inflected disease. With the help of Caputo fractional derivative, We capture the epidemiological system for the transmission of Novel Coronavirus-19 Infectious Disease (nCOVID-19). We constructed the model in four compartments susceptible, exposed, infected and recovered. We obtained the conditions for existence and Ulam’s type stability for proposed system by using the tools of non-linear analysis. The author’s thoroughly discussed the local and global asymptotical stabilities of underling model upon the disease free, endemic equilibrium and reproductive number. We used the techniques of Laplace Adomian decomposition method for the approximate solution of consider system. Furthermore, author’s interpret the dynamics of proposed system graphically via Mathematica, from which we observed that disease can be either controlled to a large extent or eliminate, if transmission rate is reduced and increase the rate of treatment.

Topological indices are numeric values associated with a graph and characterize its structure. There are various topological indices in graph theory such as degree‐based, distance‐based, and counting‐related topological indices. Among these indices, degree‐based indices are very interesting and studied well in literature. In this work, we studied the generalized form of harmonic, geometric‐arithmetic, Kulli–Basava indices, and generalized power‐sum‐connectivity index for special graph that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, wheel graph, gear graph, helm graph, and square lattice graph. We found exact values for the stated indices and for the stated special graphs. We also investigated the generalized form of the indices for various properties of alkane isomers, from which we obtained interesting results which are closed to that of experimental obtained results.

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for k-generalized quasi-trees where optimal graphs were found. Further, in this work, we extend the idea of optimization to families of graphs, including uni-cyclic graphs, bi-cyclic graphs and trees. The optimization is carried out by means of operations named as Operation A, B, C and D. The first two operations increase the value of the general power-sum connectivity index, while the last two work opposite to Operations A and B. These operations are explained by means of diagrams, where one can easily obtain their working procedures.

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is $\mathsf{\Omega }$$Y_{\alpha }(\mathsf{\Omega })={\sum }_{\zeta \varrho \in E(\mathsf{\Omega })}{(d_{\mathsf{\Omega }}{\left(\zeta \right)}^{d_{\mathsf{\Omega }}\left(\zeta \right)}+d_{\mathsf{\Omega }}{\left(\varrho \right)}^{d_{\mathsf{\Omega }}\left(\varrho \right)})}^{\alpha }$, while the second form of the GPSCI is defined as $Y_{\beta }(\mathsf{\Omega })={\sum }_{\zeta \varrho \in E(\mathsf{\Omega })}{(d_{\mathsf{\Omega }}{\left(\zeta \right)}^{d_{\mathsf{\Omega }}\left(\zeta \right)}\times d_{\mathsf{\Omega }}{\left(\varrho \right)}^{d_{\mathsf{\Omega }}\left(\varrho \right)})}^{\beta }$. In this paper, we investigate $Y_{\beta }$ in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for $Y_{\beta }$ using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered.

Let be a graph having no loop or multiple edges, order vertex partition for is represented by . The vector is the representation of vertex with respect to . If the representation of all the vertices with respect to is different, then is said to be resolving partition for the graph . The minimum number is resolving partition for and is termed as partition dimension for , represented by . There are numerous applications of partition dimension in different fields such as optimization, computer, mastermind games, and networking and also in modeling of numerical structure. The problem of finding constant value of partition dimension for a graph or network is very hard, so one can find bounds for the partition dimension. In this work, we consider convex polytopes in their generalized forms that are