We present some results on the growth in various products of graphs. In particular we study the Cartesian, strong, lexicographic, tensor and free product of graphs. We show that with respect to distances the tensor product behaves differently from other products. In general the results are valid for rooted graphs but have especially nice structure in the case of vertex-transitive factors.

The rupture degree of a noncomplete-connected graph G is defined by , where the number components and order largest component of. In this paper, we determine some Cartesian product graphs.

We extend two description logics by introducing cartesian product (CP) of concepts and roles. In SROIQ , we show how CP axioms can be reduced using other language constructs. We present a polynomial algorithm for subsumption checking in EL++ with CP axioms. We prove that the introduction of CP does not increase the complexity of reasoning tasks in both SROIQ and EL++ .

We show that every simple, (weakly) connected, possibly directed and infinite, hypergraph has a unique prime factor decomposition with respect to the (weak) Cartesian product, even if it has infinitely many factors. This generalizes previous results for graphs and undirected hypergraphs to directed and infinite hypergraphs. The proof adopts the strategy outlined by Imrich and Žerovnik for the c...