Abstract: We showed that: If $G$ is a non-bipartite connected graph, then $q(G)\geqslant \frac{1}{n(D+1)}$, where $g(G)$ is the least $Q$-eigenvalue of $G$, and $D$ is the diameter of $G$. Some relations between the least $Q$-eigenvalue of $G$ and that of its subgraph were given.

Key words: non-bipartite graph, $Q$-eigenvalue, diameter