Spectral extremal results on edge blow-up of graphs

Abstract

The edge blow-up $F^{p+1}$ of a graph $F$ for an integer $p\geq 2$ is obtained by replacing each edge in $F$ with a $K_{p+1}$ containing the edge, where the new vertices of $K_{p+1}$ are all distinct. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. In this paper, we determine the range of $spex(n,F^{p+1})$ when $F$ is bipartite and the exact value of $spex(n,F^{p+1})$ when $F$ is non-bipartite for sufficiently large $n$, which are the spectral versions of Tur\'{a}n's problems on $ex(n,F^{p+1})$ solved by Yuan [J. Combin. Theory Ser. B 152 (2022) 379--398]. This generalizes several previous results on $F^{p+1}$ for $F$ being a matching, or a star. Additionally, we also give some other interesting results on $F^{p+1}$ for $F$ being a path, a cycle, or a complete graph. To obtain the aforementioned spectral results, we utilize a combination of the spectral version of the Stability Lemma and structural analyses. These approaches and tools give a new exploration of spectral extremal problems on non-bipartite graphs