Local properties of the spectral radius and Perron vector in graphs

In 2002, Nikiforov proved that for an n-vertex graph G with clique number ω and edge number m, its spectral radius $\lambda \left(G\right)$ satisfies $\lambda \left(G\right)\le \sqrt{2(1-1/\omega )m}$, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, showing that ${\lambda }^2\left(G\right)\le 2{\sum }_{e\in E\left(G\right)}\frac{c\left(e\right)-1}{c\left(e\right)}$, where $c\left(e\right)$ is the order of the largest clique containing the edge e in G. We also characterize the extremal graphs. Furthermore, we prove that our theorem implies Nikiforov's theorem and provide an example in which the difference of Nikiforov's bound and ours is $\Omega \left(\sqrt{m}\right)$ for some cases. Our second result explores local properties of the Perron vector of graphs. 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.