2 0 Ju l 1 99 9 When is a Tensor Product of Circulant Graphs Circulant ?

The tensor product, also called direct, categorical, or Kronecker product of graphs, is one of the least-understood graph products. In this paper we determine partial answers to the question given in the title, thereby significantly extending results of Broere and Hattingh (see [2]). We characterize completely those pairs of complete graphs whose tensor products are circulant. We establish that if the orders of these circulant graphs have greatest common divisor of 2, the product is circulant whenever both graphs are bipartite. We also establish that it is possible for one of the two graphs not to be circulant and the product still to be circulant. Throughout this paper, we will assume graphs to be connected unless otherwise stated, and without multiple edges. However, we will permit graphs to have loops at some vertices. The tensor product G ⊗ H of two graphs G and H is the graph with vertex set V (G) × V (H) and edge set consisting of those pairs of vertices (g, h), (g ′ , h ′) where g is adjacent to g ′ and h is adjacent to h ′. This product, also called the Kronecker product, weak product, direct product, categorical product, and conjunction, has been studied for decades. Basic properties of the tensor product may be found, for example, in [11, 6, 9, 10]. Despite this study, many basic properties of the product are unknown or only partly understood. In this paper we determine when the tensor product 1