Expected Value of Zagreb Indices of Random Bipartite Graphs

‎In this paper‎, ‎we calculate the expected values of the first and second Zagreb indices‎, ‎denoted as $\textbf{E}\left(M_1\right)$ and $\textbf{E}\left(M_2\right)$ respectively‎, ‎as well as the expected value of the forgotten index‎, ‎$\textbf{E}\left(F\right)$‎, ‎for two models of random bipartite graphs‎. ‎To evaluate our findings‎, ‎we establish the growth rate by demonstrating that for a random bipartite graph $G$ of order $n$ in either model‎, ‎the expected value of $M_1(G)$ is $O\left( n^3 \right)$‎. ‎Furthermore‎, ‎we prove that the expected values of $M_2(G)$ and $F(G)$ are both $O\left( n^4 \right)$‎.