A Brualdi–Hoffman–Turán problem on cycles

Brualdi–Hoffman–Turán-type problem asks what is the maximum spectral radius ${\lambda }_1\left(G\right)$ of an $H$-free graph $G$ on $m$ edges? This problem gives a spectral perspective on the existence of a subgraph $H$. A significant result, due to Nikiforov, states that ${\lambda }_1\left(G\right)\le \sqrt{2m(1-\frac{1}{r})}$ for every $K_{r+1}$-free graph $G$ (Nikiforov, 2002). Bollobás and Nikiforov further conjectured ${\lambda }_1^2\left(G\right)+{\lambda }_2^2\left(G\right)\le 2m(1-\frac{1}{r})$ for every $K_{r+1}$-free graph $G$ (Bollobás and Nikiforov, 2007). Let $C_k^{+}$ denote the graph obtained from a $k$-cycle by adding a chord between two vertices of distance two. Zhai, Lin and Shu conjectured that for $k\ge 2$ and $m$ sufficiently large, if $G$ is a $C_{2k+1}$-free or $C_{2k+2}$-free graph, then ${\lambda }_1\left(G\right)\le \frac{k-1+\sqrt{4m-k^2+1}}{2}$, with equality if and only if $G\cong K_k\nabla (\frac{m}{k}-\frac{k-1}{2})K_1$ (Zhai et al., 2021). This conjecture was also included in a survey of Liu and Ning as one of twenty unsolved problems in spectral graph theory. Recently, Y.T. Li posed a stronger conjecture, which states that the above spectral bound holds for $C_{2k+1}^{+}$-free and $C_{2k+2}^{+}$-free graphs. In this paper, we confirm these two conjectures by using $k$-core method and spectral techniques. This presents a new approach to study Brualdi–Hoffman–Turán problems