de Freitas–Nikiforov–Patuzzi conjecture on the Q-index of graphs forbidding even cycles

Let $G$ be a graph of order $n$, and let $q(G)$ be the largest eigenvalue of its signless Laplacian. Let $S_{n,k}=K_k\vee\overline{K}_{n-k}$, and let $S_{n,k}^{+}$ be obtained from $S_{n,k}$ by adding an edge.

de Freitas–Nikiforov–Patuzzi conjecture. For $k\geq 2$ and a graph $G$ of sufficiently large order $n$, if $G$ has no cycle of length $2k+1$, then

$$q(G)<q(S_{n,k}),$$

unless $G=S_{n,k}$. If $G$ has no cycle of length $2k+2$, then

$$q(G)<q(S_{n,k}^{+}),$$

unless $G=S_{n,k}^{+}$.

The paper states that its results complete the proof of this conjecture, so both assertions are solved.