A graph G is an efficient open domination graph if there exists a subset D of V (G) for which the open neighborhoods centered in vertices of D form a partition of V (G). We completely describe efficient domination graphs among direct, lexicographic and strong products of graphs. For the Cartesian product we give a characterization when one factor is K2 and some partial results for grids, cylind...

The group property FW stands in-between the celebrated Kazhdan’s (T) and Serre’s FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended. It follows from work of Y. Cornulier that a wreath product G≀XH has if only both G H have X is finite. aim this paper to give an elementary, direct explicit proof fact using graphs.

The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...