Abstract

AbstractIn 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 G1 and G2, if V(H) = V(G1) × V(G2) 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. © 2011 Wiley Periodicals, Inc. J Graph Theory

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