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arXiv · 2605.30821

The spectral inducibility of graphs

Liying Kang·Xizhi Liu·Yongchun Lu
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Abstract

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 $\alpha$-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 $\alpha\to\infty$, 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 $\alpha\ge1$, 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 $r\le 2^a-1$.

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Liying Kang, Xizhi Liu, Yongchun Lu. 2026-05-29. The spectral inducibility of graphs. https://arxiv.org/abs/2605.30821

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