Spectral Turán problem of non-bipartite graphs: Forbidden books

Abstract

A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\ge 0$ is an integer. Zhai and Lin (2023) proved that for $n\ge \frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then $\rho \left(G\right)\le \rho \left(T_{n,2}\right)$, with equality if and only if $G\cong T_{n,2}$. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant result, we investigate the spectral Turán problem of non-bipartite $B_{r+1}$-free graphs of order $n$. For $r=0$, Lin et al. (2021) provided a complete solution and proved a nice result: If $G$ is a non-bipartite triangle-free graph of order $n$, then $\rho \left(G\right)\le \rho \left(S{K}_{\lfloor \frac{n-1}{2}\rfloor ,\lceil \frac{n-1}{2}\rceil }\right)$, with equality if and only if $G\cong S{K}_{\lfloor \frac{n-1}{2}\rfloor ,\lceil \frac{n-1}{2}\rceil }$, where $S{K}_{\lfloor \frac{n-1}{2}\rfloor ,\lceil \frac{n-1}{2}\rceil }$ is the graph obtained from $K_{\lfloor \frac{n-1}{2}\rfloor ,\lceil \frac{n-1}{2}\rceil }$ by subdividing an edge.

For general $r\ge 1$, let $K_{\lfloor \frac{n-1}{2}\rfloor ,\lceil \frac{n-1}{2}\rceil }^{r,r}$ be the graph obtained from $K_{\lceil \frac{n-1}{2}\rceil ,\lfloor \frac{n-1}{2}\rfloor }$ by adding a new vertex $v_0$ such that $v_0$ has exactly $r$ neighbors in each part of $K_{\lceil \frac{n-1}{2}\rceil ,\lfloor \frac{n-1}{2}\rfloor }$. By adopting a different technique named the residual index, Chvátal-Hanson theorem and typical spectral extremal methods, we in this paper prove that: If $G$ is a non-bipartite $B_{r+1}$-free graph of order $n$, then $\rho \left(G\right)\le \rho \left(K_{\lfloor \frac{n-1}{2}\rfloor ,\lceil \frac{n-1}{2}\rceil }^{r,r}\right)$ , with equality if and only if $G\cong K_{\lfloor \frac{n-1}{2}\rfloor ,\lceil \frac{n-1}{2}\rceil }^{r,r}$. An interesting phenomenon is that the spectral extremal graphs are completely different for $r=0$ and general $r\ge 1$.