  In this paper we study a new product of graphs called {\em tight product}. A
graph $H$ is said to be a tight product of two (undirected multi) graphs $G_1$
and $G_2$, if $V(H)=V(G_1)\times V(G_2)$ and both projection maps $V(H)\to
V(G_1)$ and $V(H)\to V(G_2)$ are covering maps. It is not a priori clear when
two given graphs have a tight product (in fact, it is $NP$-hard to decide). We
investigate the conditions under which this is possible. This perspective
yields a new characterization of class-1 $(2k+1)$-regular graphs. We also
obtain a new model of random $d$-regular graphs whose second eigenvalue is
almost surely at most $O(d^{3/4})$. This construction resembles random graph
lifts, but requires fewer random bits.
