Eigenvalues of the adjacency matrix of tetrahedral graphs

Abstract

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. Ifn 2 (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)n 2 (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].