Suppose I have some $d$-regular graph $G$. Let $\lambda = \max\{\lambda_2(G), |\lambda_n(G)|\}$ be the second largest eigenvalue of the adjacency matrix of $G$. Now take $\tilde{G}$, the suspension of $G$, obtained by adding a new vertex $v$ and connecting $v$ to all vertices of $G$. Is it true that $\lambda(\tilde{G}) \leq \lambda(G)$? (or more generally, is it true if instead of suspending with one vertex we suspend with a clique of size $n$)

The intuition is that $\tilde{G}$ should exhibit better mixing (hence, smaller $\lambda$), since it's easier for a random walk to get from one vertex to any other.