Let $A_1$ and $A_2$ be adjacency matrices of two nonsingular bipartite graphs of order $m$ and $n$.
Let $C_1=\begin{pmatrix} 0 & 0 &0 & \cdots 0\\ \vdots&\vdots&\vdots&\vdots \\ 0 & 0 &0 & \cdots 0\\ 1 & 1 &0 & \cdots 0\\ 1 & 1 &0 & \cdots 0\\ 0 & 0 &0 & \cdots 0\\ \end{pmatrix}$ and $C_2=\begin{pmatrix} 0 & 0 &0 & \cdots 0\\ \vdots&\vdots&\vdots&\vdots \\ 0 & 0 &0 & \cdots 0\\ 1 & 1 &0 & \cdots 0\\ \end{pmatrix}$ be two $m\times n$ matrices.
Define $A=\begin{pmatrix} A_1 & C_1\\ C_1^T & A_2 \end{pmatrix} \text{ and } B=\begin{pmatrix} A_1 & C_2\\ C_2^T & A_2 \end{pmatrix}$.
Is it true that the largest eigenvalue of $A$ is greater than that of $B$? Any hint to prove this?