Survey of Double Vertex Graphs

Abstract.

 Let G be a (V,E) graph of order p≥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}∩{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.