Some Characteristics of Double Vertex Graph

The double vertex graph 𝑈2(𝐺) of a graph 𝐺 of order 𝑛 ≥ 2 is the graph whose vertex set 
consists of all (𝑛2) unordered pairs of vertices of 𝐺 and where two vertices {𝑎, 𝑏} and {𝑐, 𝑑} are 
adjacent if and only if |{𝑎, 𝑏} ∩ {𝑐, 𝑑}| = 1 and if 𝑎 = 𝑐, then 𝑏 and 𝑑 are adjacent in 𝐺. In this 
paper, we discuss some properties of 𝑈2(𝐺) related to degree and distances. We also have 
obtained graphs 𝐺 whose double vertex graphs contain 𝐺 as a subgraph.