Chemical graph theory plays an important role in modeling and designing any chemical structure. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper, we study the chemical graph of the crystal structure of titanium difluoride $Ti{F}_2$ and the crystallographic structure of cuprite $C{u}_{2}O$. Furthermore, we compute degree-based topological indices, mainly $ABC$, $GA$, $AB{C}_4$, $G{A}_5$ and general Randić indices. Furthermore, we also give exact results of these indices for the crystal structure of titanium difluoride $Ti{F}_2$ and the crystallographic structure of cuprite $C{u}_{2}O$.

The practical and theoretical significance of the resolvability parameter makes it an important factor, particularly in the context of network analysis. Its significance is seen in various applications and consequences: Network security, efficient routing, social network analysis, facility location, and site selection. This article finds the double resolvability parameters of the fosmidomycin anti-malaria drug. Resolvability parameters like double metric, double edge metric, and double mixed metric dimensions of fosmidomycin anti-malaria drug also hold exchange properties in the molecular graph of fosmidomycin. We convert the molecular structures of fosmidomycin into molecular graphs and then find some resolvability parameters.

The advancement in coloring schemes of graphs is expanding over time to solve emerging problems. Recently, a new form of coloring, namely $P_3$-coloring, was introduced. A simple graph is called a $P_3$-colorable graph if its vertices can be colored so that all the vertices in each $P_3$ path of the graph have different colors; this is called the $P_3$-coloring of the graph. The minimum number of colors required to form a $P_3$-coloring of a graph is called the $P_3$-chromatic number of the graph. The aim of this article is to determine the $P_3$-chromatic number of different well-known classes of bipartite graphs such as complete bipartite graphs, tree graphs, grid graphs, and some special types of bipartite graphs. Moreover, we have also presented some algorithms to produce a $P_3$-coloring of these classes with a minimum number of colors required.