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

Nikiforov's well-known spectral Tur\'an inequality for walks states that, for every graph $G$ with clique number $\omega(G)$, $\lambda^r(G)\le w_r(G)(1-1/\omega(G))$, where $\lambda(G)$ is the largest eigenvalue of the adjacency matrix of $G$, and $w_r(G)$ is the number of walks with $r$ vertices in $G$. For $r=1$, this is Wilf's inequality; for $r=2$, it gives Nikiforov's spectral Tur\'an theorem. Recently, Liu and Ning proved local versions of these two inequalities, strengthening both Wilf's inequality and Nikiforov's spectral Tur\'an theorem. It is natural to ask whether Nikiforov's spectral Tur\'an inequality for walks also admits a local strengthening. Motivated by this question, Kannan, Kumar, and Pragada conjectured the vertex-local bound $\lambda^r(G)\le \sum_{v\in V(G)} w_r(v)(1-1/c_G(v))$, where $w_r(v)$ denotes the number of walks with $r$ vertices starting at $v$, and $c_G(v)$ is the maximum order of a clique containing $v$. This conjecture is important because it gives the most natural local form of Nikiforov's spectral Tur\'an inequality for walks. In this paper, we confirm this conjecture. More precisely, for $r\ge 2$, we prove the stronger edge-local inequality $$\lambda^r(G) \le \sum_{uv\in E(G)} \frac{c_G(uv)-1}{c_G(uv)} \bigl(w_{r-1}(u)+w_{r-1}(v)\bigr),$$ where $c_G(uv)$ is the maximum order of a clique containing the edge $uv$. Our result implies Nikiforov's spectral Tur\'an inequality for walks and unifies several local spectral extremal results of Liu and Ning. We also determine all extremal graphs for both the edge-local and vertex-local inequalities. The main new ingredient is a Markov-chain estimate whose transition matrix is constructed from a Perron vector of $A(G)$; this estimate carries the local edge coefficient through walks of arbitrary length.