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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.30821 |
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| _version_ | 1866910271636766720 |
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| author | Kang, Liying Liu, Xizhi Lu, Yongchun |
| author_facet | Kang, Liying Liu, Xizhi Lu, Yongchun |
| contents | We introduce a spectral version of the classical inducibility problem. Given an $\ell$-vertex graph $F$ and an $n$-vertex graph $G$, let $H_F(G)$ be the $\ell$-uniform hypergraph whose edges are the $\ell$-sets inducing a copy of $F$ in $G$. We study the maximum possible $α$-spectral radius of $H_F(G)$ over all $n$-vertex graphs $G$. For fixed $G$, this spectral parameter tends to $\ell!$ times the number of induced copies of $F$ in $G$ as $α\to\infty$, and therefore refines the usual induced-copy count.
Our main result is a spectral analogue of the Brown--Sidorenko reduction: for every complete multipartite graph $F$, every $n$, and every $α\ge1$, a spectral extremal graph can be chosen to be complete multipartite. We also show that the leading asymptotic constant is the ordinary inducibility $i(F)$, and obtain exact multipartite reductions for stars $K_{1,t}$ and balanced complete $r$-partite graphs $K_{a,\ldots,a}$ with $r\le 2^a-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30821 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The spectral inducibility of graphs Kang, Liying Liu, Xizhi Lu, Yongchun Combinatorics We introduce a spectral version of the classical inducibility problem. Given an $\ell$-vertex graph $F$ and an $n$-vertex graph $G$, let $H_F(G)$ be the $\ell$-uniform hypergraph whose edges are the $\ell$-sets inducing a copy of $F$ in $G$. We study the maximum possible $α$-spectral radius of $H_F(G)$ over all $n$-vertex graphs $G$. For fixed $G$, this spectral parameter tends to $\ell!$ times the number of induced copies of $F$ in $G$ as $α\to\infty$, and therefore refines the usual induced-copy count. Our main result is a spectral analogue of the Brown--Sidorenko reduction: for every complete multipartite graph $F$, every $n$, and every $α\ge1$, a spectral extremal graph can be chosen to be complete multipartite. We also show that the leading asymptotic constant is the ordinary inducibility $i(F)$, and obtain exact multipartite reductions for stars $K_{1,t}$ and balanced complete $r$-partite graphs $K_{a,\ldots,a}$ with $r\le 2^a-1$. |
| title | The spectral inducibility of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.30821 |