A tetrahedral graph is defined to be a graphG, whose vertices are identified with the\(\left( {\begin{array}{*{20}c} n \\ 3 \\ \end{array} } \right)\) unordered triplets onn symbols, such that vertices are adjacent if and only if the corresponding triplets have two symbols in common. Ifn2(x) denotes the number of verticesy, which are at distance 2 fromx andA(G) denotes the adjacency matrix ofG, thenG has the following properties: P1) the number of vertices is\(\left( {\begin{array}{*{20}c} n \\ 3 \\ \end{array} } \right)\). P2)G is connected and regular. P3)n2(x) = 3/2(n−3)(n−4) for allx inG. P4) the distinct eigenvalues ofA(G) are −3, 2n−9,n−7, 3(n−3). We show that, ifn > 16, then any graphG (with no loops and multiple edges) having the properties P1)–P4) must be a tetrahedral graph. An alternative characterization of tetrahedral graphs has been given by the authors in [1].