Survey of double vertex graphs

Let G be a (V, E) graph of order p greater than or equal to 2. The double vertex graph U(2)(G) is the graph whose vertex set consists of all 2-subsets of V such that two distinct vertices {x,y} and {u,v} are adjacent if and only if\{x,y} boolean AND {u, v}\ = 1 and if x = u, then y and v are adjacent in G. For this class of graphs we discuss the regularity, eulerian, hamiltonian, and bipartite properties of these graphs. A generalization of this concept is n-tuple vertex graphs, defined in a manner similar to double vertex graphs. We also review several recent results for n-tuple vertex graphs.