Nickel(II) porphyrins typically adopt a square planar coordination geometry, with the nickel atom located at the center of the porphyrin ring and the coordinating atoms arranged in a square plane. The additional atoms or groups coordinated to the nickel atom in nickel(II) porphyrins are called ligands. Porphyrins have been investigated as potential agents for imaging and treating cancer due to their ability to selectively bind to tumor cells and be used as sensors for a variety of analytes. Nickel(II) porphyrins are relatively stable compounds, with high thermal and chemical stability. They can be stored in a solid state or in solution without significant degradation. In this study, we compute several connectivity indices, such as general Randi’c, hyper Zagreb, and redefined Zagreb indices, based on the degrees of vertices of the chemical graph of nickel porphyrins. Then, we compute the entropy and heat of formation NiP production, among other physical parameters. Using MATLAB, we fit curves between various indices and the thermodynamic properties parameters, notably the heat of formation and entropy, using various linearity- and non-linearity-based approaches. The method’s effectiveness is evaluated using $$R^2$$R2 , the sum of squared errors, and root mean square error. We also provide visual representations of these indexes. These mathematical frameworks might offer a mechanism to investigate the thermodynamical characteristics of NiP’s chemical structure under various circumstances, which will help us understand the connection between system dimensions and these metrics.

A $k$- ${\gamma }_c$-critical graph is a graph $G$ with connected domination number ${\gamma }_c\left(G\right)=k$ and ${\gamma }_c(G+uv)<k$ for any pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\omega$ and $\alpha$ be respectively the clique number and the independence number of a graph. In this paper, we prove that every $k$- ${\gamma }_c$-critical graph satisfies $\alpha +\omega \le n-\lfloor \frac{k}{2}\rfloor +1$ for $1\le k\le 3$. We also characterize all $3$- ${\gamma }_c$-critical graphs achieving the upper bound. For $k\ge 4$, we show that there are infinitely many $k$- ${\gamma }_c$-critical graphs satisfying $\alpha +\omega =n-\lfloor \frac{k}{2}\rfloor +1$. Thus, we conclude this paper with an open problem that every $k$- ${\gamma }_c$-critical graph for $k\ge 4$ satisfies $\alpha +\omega \le n-\lfloor \frac{k}{2}\rfloor +1$.

A Cop and Drunken Robber (CDR) game is one variation of a famous combinatorial game, called Cops and Robbers, which has been extensively studied and applied in the area of theoretical and computer science as demonstrated by several conferences and publications. In this paper, for a natural number n, we present two strategies for a single cop to chase a drunken robber on n-dimensional infinite-grid graphs. Both strategies show that if the initial distance between the cop and the drunken robber is s, then the expected capture time is $s+o\left(s\right)$.