In this article, we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G 1 and G2, if V(H) = V(G1) × V(G 2) and both projection maps V(H)→V(G1) and V(H)→V(G2) 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(d3/4). This construction resembles random graph lifts, but requires fewer random bits.
- lifts of graphs
- random graphs