K e y w o r d s D i s t a n c e r e g u l a r graphs, Direct product. 1. I N T R O D U C T I O N E v e r y c o n n e c t e d g r a p h is r e p r e s e n t a b l e by m e a n s o f a level diagram (cf. [1]) as follows. C h o o s e a v e r t e x u, a n d le t i t be t h e sole r e s iden t of level zero. T h e ve r t i c e s on level i a re p rec i se ly t h o s e w h o s e d i s t a n c e f r o...

The planarity of the direct product of two graphs has been widely studied in the past. Surprisingly, the missing part is the product with K2, which seems to be less predictible. In this piece of work, we characterize which subdivisions of multipartite complete graphs, have their direct product with K2 planar. This can be seen as a step towards the characterization of all such graphs.

In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs G1 and G2 are equal to κ1v2 and min{λ1v 2 2 , δ2 +δ1v2}, respectively, where δi, κi, λi and vi denote the minimum degree, the connectivity, the edge-connectivity and the number of vertices of Gi, respectively. We also obtain that the edge-connectivity of the direct product of K2 an...

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...

in this paper, the weighted szeged indices of cartesian product and corona product of twoconnected graphs are obtained. using the results obtained here, the weighted szeged indices ofthe hypercube of dimension n, hamming graph, c4 nanotubes, nanotorus, grid, t− fold bristled,sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.

Let G = (V,E) be a graph of order n. A distance magic labeling of G is a bijection l : V → {1, . . . , n} for which there exists a positive integer k such that ∑ x∈N(v) l(x) = k for all v ∈ V , where N(v) is the neighborhood of v. We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second f...