Maxima of the Q-index for 3K₃-free graphs

The Q-index of a graph G is the largest eigenvalue of its Q-matrix Q(G)=D(G)+A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let 3K₃ denote the graph consisting of three vertex-disjoint triangles. A graph is called 3K₃-free if it does not contain 3K₃ as a subgraph. In this paper, we present a sharp upper bound on the Q-index of 3K₃-free graphs of order n≥453, and characterize the unique extremal graph which attains the bound.