Nikiforov's spectral consecutive cycle problem and the connected-matching method

Abstract

Let $\ddot{I}(G)$ denote the adjacency spectral radius of a graph $G$ of order $n$. We determine the sharp constant in an open problem of Nikiforov (2008) on cycles of consecutive lengths. For every $\epsilon >0$ and all sufficiently large $n$, if $G$ is an $n$-vertex graph with $\ddot{I}(G)>\sqrt{\lfloor n^2/4\rfloor },$ then $G$ contains a cycle $C_{\ell }$ for every integer length $3\le \ell \le (\frac{3-\sqrt{5}}{2}-\epsilon )n.$ The constant $(3-\sqrt{5})/2$ is best possible, as shown by the split graph $K_k\vee {\overline{K}}_{n-k}$ with $k\sim (3-\sqrt{5})n/4$. Our result improves all previous results [LAA2008, CPC2020, JGT2023, JGT2023, GC2024]. The proof combines the degree form of Szemerédi's regularity lemma, a spectral matching theorem of Feng-Yu-Zhang, Weyl's inequality, a refinement of Łuczak's connected-matching embedding method, and other ideas.

21 pages