  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
$\lambda (G) \leq \sqrt{2(1 - 1/\omega) m}$, which confirmed a conjecture
implicitly suggested by Edwards and Elphick. In this paper, we prove a local
version of spectral Tur\'an inequality, which states that $\lambda^2(G)\leq
2\sum_{e\in E(G)}\frac{c(e)-1}{c(e)}$, 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 $\Omega
(\sqrt{m})$ 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 Tur\'an 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 $k\geq 2$ and
an independent set $S$, we have \[ \sum_{v\in S} x_v^2 \leq \frac{1}{2} -
\frac{k-2}{2\sqrt{(k-2)^2 + 4(k-1)(n-k+1)}}, \] 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.
