Zig-zag product

For a port-labeled $D$-regular graph $G$ and a $d$-regular graph $H$ on its $D$ port labels, replace every vertex by a copy of $H$. An edge makes a step in $H$, crosses the corresponding edge of $G$, then makes another step in $H$. The result has $|G|D$ vertices and degree $d^2$. Its normalized operator is $BRB$, where $B=I\otimes P_H$ and $R$ is the involutive edge rotation of $G$. Decomposing a mean-zero vector into cloud-constant and cloud-orthogonal parts proves the bound $\rho (G\mathrm{zigzag}H)\le \rho (G)+2\rho (H)+\rho (H{)}^2$.

Choose $D=d^2$ and a fixed $d$-regular graph $H$ on $D^2$ vertices with absolute nonconstant eigenvalues at most $1/10$. Start with the loop-inclusive complete graph $G_0$ on $D$ vertices and iterate $G_{t+1}=G_t^2\mathrm{zigzag}H$. The degree stays $D$, vertex counts are $D^{2t+1}$, and ${\rho }_{t+1}\le {\rho }_t^2+21/100$. Since ${\rho }_0=0$, all ${\rho }_t\le 1/2$. The spectral cut bound gives multiplicity-counted edge expansion at least $D/4$. Removing loops and merging parallel edges loses at most a factor $D$, producing simple expander graphs with maximum degree $D$ and edge expansion at least $1/4$.