On the maximum Zagreb indices of bipartite graphs with given connectivity

The first Zagreb index $M_1$ of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index $M_2$ of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices. In this paper, we study the Zagreb indices of bipartite graphs of order $n$ with $\kappa (G)=k$ (resp. ${\kappa }^{\mathrm{\prime }}(G)=s$) and sharp upper bounds are obtained for $M_1(G)$ and $M_2(G)$ for $G\in {\mathcal{V}}_n^k$ (resp. ${\mathcal{E}}_n^s$), where ${\mathcal{V}}_n^k$ is the set of bipartite graphs of order $n$ with $\kappa (G)=k$, and ${\mathcal{E}}_n^s$ is the set of bipartite graphs of order $n$ with ${\kappa }^{\mathrm{\prime }}(G)=s$.