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

Abstract

The $Q$-index of a graph $G$ is the largest eigenvalue of its $Q$-matrix $Q\left(G\right)=D\left(G\right)+A\left(G\right)$, where $D\left(G\right)$ and $A\left(G\right)$ are the diagonal matrix of vertex degrees and the adjacency matrix of $G$, respectively. Let $3K_3$ denote the graph consisting of three vertex-disjoint triangles. A graph is called $3K_3$-free if it does not contain $3K_3$ as a subgraph. In this paper, we present a sharp upper bound on the $Q$-index of $3K_3$-free graphs of order $n\ge 453$, and characterize the unique extremal graph which attains the bound.