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Abstract:Nikiforov conjectured that, for every fixed $k\ge2$ and all sufficiently large $n$, the unique $n$-vertex $C_{2k+2}$-free graph with maximum adjacency spectral radius is $S^+_{n,k}$, where $S_{n,k}=K_k\vee\overline K_{n-k}$ and $S^+_{n,k}$ is obtained from $S_{n,k}$ by adding one edge inside the independent part. Cioabă, Desai and Tait proved this conjecture for $n\ge k^{O(k)}$. Later, Li and Ning raised the problem of determining the optimal exponent $\gamma=\gamma(k)$ such that the same conclusion holds for $n\ge \Omega(k^{\gamma(k)})$.
We prove a stronger uniform theorem for Nikiforov's matrices $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. More precisely, for every $\epsilon>0$ there are constants $C_\epsilon$ and $k_\epsilon$ such that for all $0\le\alpha\le1-\epsilon$, $k\ge k_\epsilon$ and $n\ge C_\epsilon k$, every $n$-vertex $C_{2k+2}$-free graph $G$ satisfies $\rho_\alpha(G)\le\rho_\alpha(S^+_{n,k})$, with equality if and only if $G\cong S^+_{n,k}$. In particular, the case $\alpha=0$ answers the problem of Li and Ning, and the $A_\alpha$-spectral even-cycle threshold is linear in $k$, uniformly for all $\alpha$ bounded away from $1$.
Our proof introduces a weighted rooted Erdős--Gallai type path lemma, which may be of independent interest in Perron-vector methods for spectral extremal graph problems. The same method also yields asymptotically tight $A_\alpha$-spectral bounds for two local forbidden-subgraph families, namely $(K_1\vee P_\ell)$-free graphs and $F_s$-free graphs, where $F_s$ denotes the friendship graph.

Submission history

From: Jiasheng Zeng [view email]
[v1] Fri, 5 Jun 2026 07:25:27 UTC (26 KB)
[v2] Sat, 22 Aug 2026 16:17:08 UTC (28 KB)