Proof of a conjecture involving the second largest D-eigenvalue and the number of triangles

Let G be a connected graph of order n with tr triangles and D be the distance matrix of G. Let λ1(D)≥λ2(D)≥⋯≥λn(D) be the D-eigenvalue of the graph G. Fajtlowicz (1998) [4] conjectured that λ2(D)≤tr when the independent number α(G)≤2. In this paper, the conjecture is confirmed and the extremal graph when the equality holds is characterized.