The number of edges of the edge polytope of a finite simple graph

Let d  ≥ 3 be an integer. It is known that the number of edges of the edge polytope of the complete graph with d vertices is d ( d  − 1)( d  − 2) / 2 . In this paper, we study the maximum possible number μ d of edges of the edge polytope arising from finite simple graphs with d vertices. We show that μ d  =  d ( d  − 1)( d  − 2) / 2 if and only if 3 ≤  d  ≤ 14 . In addition, we study the asymptotic behavior of μ d . Tran–Ziegler gave a lower bound for μ d by constructing a random graph. We succeeded in improving this bound by constructing both a non-random graph and a random graph whose complement is bipartite.