Products of Circulant Graphs Are Metacirculant

The standard products—cartesian, lexicographic, tensor, and strong—all belong to a class of products introduced by W. Imrich and H. Izbicki (1975, Monatsh. Math.80, 277–281) and later called B-products by I. Broere and J. H. Hattingh (1990, Quaest. Math.13, 191–216) who establish that the lexicographic product of two circulant graphs is again circulant. In this paper, we establish that any B-product of two circulant graphs is always a so-called metacirculant graph with parameters that are easily described in terms of the product graphs. We also establish that any metacirculant graph with the appropriate structure is isomorphic to the B-product of a pair of circulant graphs.