The edge-bandwidth of a graph G is the bandwidth of the line graph of G. We show asymptotically tight bounds on the edge-bandwidth of two dimensional grids and tori, the product of two cliques and the n-dimensional hypercube.

We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the C∗algebras and the non-sel...

The edges of the Cartesian product of graphs G x Hare to be colored with the condition that all rectangles, i.e., K2 x K2 subgraphs, must be colored with four distinct colors. The minimum number of colors in such colorings is determined for all pairs of graphs except when G is 5-chromatic and H is 4or 5-chromatic.

If G is a graph then a subgraph H is isometric if, for every pair of vertices u, v of H, we have dH(u, v) = dG(u, v) where d is the distance function. We say a graph G is distance preserving (dp) if it has an isometric subgraph of every possible order up to the order of G. We give a necessary and sufficient condition for the lexicographic product of two graphs to be a dp graph. A graph G is seq...

A k-queue layout of a graph G consists of a linear order σ of V (G), and a partition of E(G) into k sets, each of which contains no two edges that are nested in σ. This paper studies queue layouts of graph products and powers.

Let γb(G) denote the broadcast domination number for a graph G. In [Discrete Applied Math. 154 (2006), 59–75], Dunbar et al. determined the value of γb(G), where G is the Cartesian product of two paths. In this paper, we evaluate the value of γb(G), whenever G is the strong product, the direct product and the lexicographic product of two paths.

A pebbling step on a graph consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A graph is said to be cover pebbled if every vertex has a pebble on it after a series of pebbling steps. The cover pebbling number of a graph is the minimum number of pebbles such that the graph can be cover pebbled, no matter how the pebbles are initially placed on the vert...

A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a ttone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in thr...

We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the indepen...

The graph product is an operator mixing direct and free products. It is already known that free products and direct products of automatic monoids are automatic. The main aim of this paper is to prove that graph products of automatic monoids of finite geometric type are still automatic. A similar result for prefix-automatic monoids