For each undirected (and unweighted) graph $G$ we define the Laplacian matrix by $L(G) = D(G)-A(G)$, where $A(G)$ should denote the adjacency matrix and $D(G)$ the diagonal matrix, where $D(G)_{ii}$ denotes the degree of the $i$-th vertex of $G$ (after some ordering of the vertex set). We denote the eigenvalues of $L(G)$ by $\lambda_0 = 0 \leq \lambda_1 \leq \ldots$

If we consider the $n$-dimensional hypercube graph (see http://en.wikipedia.org/wiki/Hypercube_graph for more information), we obtain that the laplacian eigenvalues are $2i$ with multiplicity $\binom{n}{i}$ for $i=0,\ldots,n$. Hence, $\lambda_1 = 2$. The graph has $n\cdot 2^{n-1}$ edges.

My question is now: Is there any method known to construct for increasing $n$ a graph $G$ with $2^n$ vertices and $n\cdot 2^{n-1}$ edges, where the $\lambda_1$ of $L(G)$ is bigger than 2 (the more the better)? And is there any estimate to these $\lambda_1$ then? Furthermore, it would interest me to consider such a kind graphes $G$, where $\lambda_1$ is still bigger than 2, but the graph itself looks quite similar to the $n$-dimensional hypercube graph, but this is only optional.