  Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex
$H$-minor free graphs. The problem on determining this extremal value can be
dated back to the early 1990s. Up to now, it has been solved for $n$
sufficiently large and some special minors, such as $\{K_{2,3},K_4\}$,
$\{K_{3,3},K_5\}$, $K_r$ and $K_{s,t}$. In this paper, we find some unified
phenomena on general minors. Every graph $G$ on $n$ vertices with spectral
radius $\rho\geq spex(n,H_{minor})$ contains either an $H$ minor or a spanning
book $K_{\gamma_H}\nabla(n-\gamma_H)K_1$, where $\gamma_H=|H|-\alpha(H)-1$.
Furthermore, assume that $G$ is $H$-minor free and $\Gamma^*_s(H)$ is the
family of $s$-vertex irreducible induced subgraphs of $H$, then $G$ minus its
$\gamma_H$ dominating vertices is $\Gamma^*_{\alpha(H)+1}(H)$-minor saturate,
and it is further edge-maximal if $\Gamma^*_{\alpha(H)+1}(H)$ is a connected
family. As applications, we obtain some known results on minors mentioned
above. We also determine the extremal values for some other minors, such as
flowers, wheels, generalized books and complete multi-partite graphs. Our
results extend some conjectures on planar graphs, outer-planar graphs and
$K_{s,t}$-minor free graphs. To obtain the results, we combine stability
method, spectral techniques and structural analyses. Especially, we give an
exploration of using absorbing method in spectral extremal problems.
