A local Tur\'an inequality for walks and the spectral radius

For a vertex $v$, let $c_G(v)$ be the order of the largest clique containing $v$, and let $w_r(v)$ be the number of walks with $r$ vertices starting at $v$. We prove that, for every finite simple graph $G$ and every integer $r\ge 1$, \begin{flalign*} \lambda_1(G)^r \le \sum_{v\in V(G)} w_r(v)\frac{c_G(v)-1}{c_G(v)}. \end{flalign*} This confirms a conjecture of Kannan, Kumar, and Pragada. It strengthens Nikiforov's walk inequality and extends, in a unified form, the localized Wilf theorem and the degree-local Tur\'an inequality of Liu and Ning. The proof is based on the stationary distribution of a Markov chain whose transition matrix is constructed from a Perron vector of $A(G)$, together with a weighted local spectral Tur\'an theorem. We determine all the extremal graphs.