ON TETRAHEDRAL GRAPH

Generalizing the concept of the triangular association scheme, Bose and Laskar introduced the tetrahedral graph the vertices of which are the unordered triplets selected from n symbols with two points adjacent if and only if their corresponding triplets have two symbols in common. If we let d(x, y) denote the distance between two vertices x, y and (x, y) the number of vertices adjacent to both x and y, then the tetrahedral graph possesses the following 4 properties:(BO) the number of vertices(Bl) it is connected and regular of degree S(n − 3)(B2) if d(x, y) = 1 then j(x, y) = n − 2(B3) if d(x, y) = 2 then j(x, y) = 4.The question whether these conditions characterize tetrahedral graphs (no loops or parallel edges permitted) was answered in the affirmative by Bose and Laskar for ^ > 16. In the present paper characterizations of tetrahedral graphs are derived by strengthening each one of (Bl), (B2), (B3) and these results are utilized to prove the sufficiency of (B0)−(B3) for n=6.(For n < 4 the problem is void, n = 4,5 are trivial cases. © 1968 by Pacific Journal of Mathematics.