A Reaserch For Given Diameter Graph About 7-distance In Planar Point Set

As an important part of scientific development, discrete and composition geometry play an important role. The paper based on g（k）（when k 37） which proposed by the famous mathematictian Erd?s and Fishburn. This paper mainly disccuss the given diameter graph about 7-distance in planar point set, and give a further reaseach for 7-distance.A planar point set X is called a k-distance sets if there are exactly k distinct distances between two points in X, the longest distance is called diameter, denoting by D. X_D denote the set of the endpoints of all diameters,m =m（X）=|X_D| define the number of elements of X_D, the diameter graphDG（X_D） composes of all diameters in X. d（v） denote the degree of vertex v in X_D. let nR denote the vertex set of a regular convex n-gon, nR -i denote a set of n -i vertices of nR. We denote a cycle with n vertices by nC. As there has been fruitful research on setting value of g（k） when k ￡6, we assume that g（k） means the number of points which belong to the largest point set provided that it has k distinct distance. This article addresses the situations of k 37, and gets some conclusion.1) We will perform an analysis the degree value d（v） of all vertices in k-distanceDG（X_D） given that 2 1Dm =X =k -, which will lead to the result that tells us d（v） ￡2.2) Based on the result, we research the situation of 7-distance. It will be153X_D =R - when the 7-distance sets DG（X_D） =P₁₀ UP₂, DG（X_D） =P₉UP₃, DG（X_D） =P₈UP₄, DG（X_D） =P₇UP₃UP₂.3) We have known the set X having at most five distances with DG（X_D） =C₇ has the unique X_D=R₇, and the set X having at most six distances with DG（X_D） =C has the unique X_D =R₉. In this paper, we prove the set X having most seven distance with DG（X_D） =C₁₁, it must has the unique X_D=R₁₁.