A local Turán inequality for walks and the spectral radius
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917457302650880 |
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| author | Liu, Feng Sun, Shuang Wang, Yan Wu, Qi |
| author_facet | Liu, Feng Sun, Shuang Wang, Yan Wu, Qi |
| contents | 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*}
λ_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 Turán 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 Turán theorem. We determine all the extremal graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02191 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A local Turán inequality for walks and the spectral radius Liu, Feng Sun, Shuang Wang, Yan Wu, Qi Combinatorics 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*} λ_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 Turán 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 Turán theorem. We determine all the extremal graphs. |
| title | A local Turán inequality for walks and the spectral radius |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.02191 |