(1) Let $G$ be a connected graph with $n = 2m$ vertices. Let $V_1$ and $V_2$ be disjoint subsets of $V(G)$ with $|V1| = |V2| = m$. Let $\mu$ be the algebraic connectivity of $G$. Show that the number of edges of $G$ with one endpoint in $V_1$ and the other in $V_2$ is at least $\frac{\mu m}{2}$. Show that equality is attained for the $n$-cube $Q_n, n \geq 2$, by taking suitable $V_1$ and $V_2$.
(2) Let $G$ be a connected graph with $n=3m$ vertices.Let $V_1,V_2,V_3$ be disjoint set of vertices such that $|V_1|=|V_2|=|V_3|=m$. Let $\mu$ be the algebraic connectivity of $G$.
Show that the number of edges of $G$ with one end-point in $V_i$ and the other end point in $V_j$ where $i\neq j$ is at least $m\mu$.
I am stuck with the above two problems. Two problems are almost same.