  The chromatic critical edge theorem of Simonovits states that for a given
color critical graph $H$ with $\chi(H)=k+1$, there exists an $n_0(H)$ such that
the Tur\'an graph $T_{n,k}$ is the only extremal graph with respect to
$ex(n,H)$ provided $n \geq n_0(H)$. Nikiforov's pioneer work on spectral graph
theory implies that the color critical edge theorem also holds if $ex(n,H)$ is
replaced by the maximum spectral radius and $n_0(H)$ is an exponential function
of $|H|$. We want to know which color critical graphs $H$ satisfy that $n_0(H)$
is a linear function of $|H|$. Previous graphs include complete graphs and odd
cycles. In this paper, we find two new classes of graphs: books and theta
graphs. Namely, we prove that every graph on $n$ vertices with
$\rho(G)>\rho(T_{n,2})$ contains a book of size greater than $\frac{n}{6.5}$.
This can be seen as a spectral version of a 1962 conjecture by Erd\H{o}s, which
states that every graph on $n$ vertices with $e(G)>e(T_{n,2})$ contains a book
of size greater than $\frac{n}{6}$. In addition, our result on theta graphs
implies that if $G$ is a graph of order $n$ with $\rho(G)>\rho(T_{n,2})$, then
$G$ contains a cycle of length $t$ for every $t\leq \frac{n}{7}$. This is
related to an open question by Nikiforov which asks to determine the maximum
$c$ such that every graph $G$ of large enough order $n$ with
$\rho(G)>\rho(T_{n,2})$ contains a cycle of length $t$ for every $t\leq cn$.

Combinatorics