Introduction. This note considers a graph product derived from the Kronecker product of matrices. Some indication of the geometrical nature of this product is given and a theorem stating necessary and sufficient conditions for a graph product to be connected is proved. The matrix analogue of the above result is also stated. I. A convenient representation for a finite undirected graph [l] G is an adjacency matrix. If the vertex set of G is {pi], i= 1, • • • , », then an adjacency matrix of G is an »X« matrix (ay) with an equal to the number of lines (paths of length one) joining p{ to p¡. A given graph is then represented by an equivalence class of matrices A = {PítIPí"1! for all permutation matrices P¿ of order equal to the order of A}. Each element of A corresponds to a different ordering of the vertices of G. It is clear that for each such class of adjacency matrices there corresponds a unique class of isomorphic graphs.From this point on "graph" will mean a finite undirected graph with no loops. Such a graph has an adjacency matrix (an) whose entries are non-negative integers such that an = an and ati = 0. We also use the following notation: o(G) is the number of vertices of G and is called the order of G, pi-^>pk is a chain in G from vertex pi to vertex pk, and n(pi->pk) is the number of lines (not necessarily dis-
A perfectdominatingset S of a graph r is a set of vertices of r such that every vertex of l' is either in S or is adjacent to exactly one vertex of S. We show that a perfect dominating set of the *cube C?, , induces a subgraph of C?,, whose components are isomorphic to hypercubes. We conjecture that each of these hypercubes has the same dimension. We then prove that if C?, is a component of the subgraph induced by S, then n -r = 1 or 3 (mod 6). A number of examples are given and connections with Steiner Systems and codes are noted.
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