The Q-index extremal conjecture for forbidden odd and even cycles

Let $G$ be a graph of order $n$. For $k\geq 2$, let $S_{n,k}=K_k\vee\overline{K}_{n-k}$, the graph obtained by joining every vertex of a complete graph of order $k$ to every vertex of an independent set of order $n-k$, and let $S_{n,k}^{+}$ be the graph obtained by adding an edge to $S_{n,k}$. Write $q(G)$ for the largest eigenvalue of the signless Laplacian of $G$. The Q-index extremal conjecture. For sufficiently large $n$, if $G$ has no cycle $C_{2k+1}$, then $q(G)<q(S_{n,k})$, unless $G=S_{n,k}$; and if $G$ has no cycle $C_{2k+2}$, then $q(G)<q(S_{n,k}^{+})$, unless $G=S_{n,k}^{+}$. This conjecture seeks the maximum signless-Laplacian eigenvalue among sufficiently large graphs avoiding a specified odd or even cycle; the paper establishes an asymptotically tight bound and describes its result as progress toward the full conjecture, which remains unresolved in the supplied text.