Shannon entropy, also known as information entropy or entropy, measures the uncertainty or randomness of probability distribution. Entropy is measured in bits, quantifying the average amount of information required to identify an event from the distribution. Shannon’s entropy theory initiates graph entropies and develops information-theoretic magnitudes for structural computational evidence of organic graphs and complex networks. Graph entropy measurements are valuable in several scientific fields, such as computing, chemistry, biology, and discrete mathematics. In this study, we investigate the entropy of fractal-type networks by considering cycle, complete, and star networks as base graphs using degree-based topological indices.

Let \(\mathbb{F}_{q}\) be a finite field, where \(q\) is an odd prime such that \(q>3\). Let \(f\left(t\right) =t^{3}-t\) \(\in \mathbb{F}_{q}\left[ t\right]\) be a polynomial of degree 3. For \(\lambda \neq 0\) in \(\mathbb{F}_{q}\), consider families of elliptic curves \(\left\{ E_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast}}\) and \(\left\{ H_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast }}\) defined respectively by\[v^{2}=\lambda f(u)\text{ and }f\left( v\right) =\lambda f(u).\]In this paper, I investigate the relation between the rational points over finite field on \(\left\{E_{\lambda }\left( \mathbb{F}_{q}\right) \right\}_{\lambda \in \mathbb{F}_{q}^{\ast }}\) and \(\left\{ H_{\lambda }\left(\mathbb{F}_{q}\right) \right\} _{_{\lambda \in \mathbb{F}_{q}^{\ast }}}\), and determine the number of rational points on both of these family of curves.