Variants of spectral Turán theorems and eigenvectors of graphs

In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $\omega$ and edge number $m$, the spectral radius $\lambda (G)$ satisfies $\lambda (G)\le \sqrt{2(1-1\mathrm{/}\omega )m}$, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, which states that ${\lambda }^2(G)\le 2{\sum }_{e\in E(G)}\frac{c(e)-1}{c(e)}$, where $c(e)$ is the order of the largest clique containing the edge $e$ in $G$. We also characterize the extremal graphs. We prove that our theorem implies Nikiforov’s theorem and give an example to show that the difference of Nikiforov’s bound and ours is $\mathrm{\Omega }(\sqrt{m})$ for some cases. Additionally, we establish a spectral counterpart to Ore’s problem (1962) which asks for the maximum size of an $n$-vertex graph such that its complement is connected and does not contain $F$ as a subgraph. Our result leads to a new spectral Turán inequality applicable to graphs with connected complements. Finally, we disprove a conjecture of Gregory, asserting that for a connected $n$-vertex graph $G$ with chromatic number $k\ge 2$ and an independent set $S$, we have $$\sum _{v\in S}{x}_v^2\le \frac{1}{2}-\frac{k-2}{2\sqrt{(k-2{)}^2+4(k-1)(n-k+1)}},$$ where $x_v$ is the component of the Perron vector of $G$ with respect to the vertex $v$. A modified version of Gregory’s conjecture is proposed.