Spectral extrema on graphs with given size forbidding small fans

Abstract

The $k$-fan $H_k$ is the graph obtained by taking the join of a vertex with a path order $k$. Brualdi-Hoffman-Turán problem asks what is the maximal spectral radius of an $H$-free graph $G$ on $m$ edges? Clearly, $H_2$ is a triangle, and $H_3$ is a book. Brualdi-Hoffman problems on triangles and books were solved by Nosal and Nikiforov. Recently, Yu, Li, and Peng showed that if $G$ is an $H_4$-free graph of size $m\geq 8$, then $\rho(G)\leq \rho(S_{\frac{m+3}{2},2})$, with equality if and only if $m$ is odd and
$G\cong S_{\frac{m+3}{2},2}$. In this note, by pure spectral techniques, we show that if $m\geq14$ and $\rho(G)\geq\frac12(1+\sqrt{4m-5})$, then $G$ contains a copy of $H_4$ unless $G\in\{S_{\frac{m+3}{2},2},S_{\frac{m+4}{2},2}^-\}$. Our result not only determines the spectral extremal value of $H_4$ for both even and odd $m$ but also implies a result on $5$-cycle-free graphs by Min, Lou, and Huang.