  Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say
that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following
properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$
for any $u,v$, where $G_{uv}$ is the graph obtained by applying Kelmans
operation from $u$ to $v$; (II) $\mathcal{P}(G) <\mathcal{P}(G+e)$ for any edge
$e\notin E(G)$. Let $P_k$ be a path of order $k$, $\mathcal{C}_{\geq k}$ the
set of all cycles of length at least $k$ and $M_{k+1}$ a matching containing
$k+1$ independent edges. In this paper, we mainly prove the following three
results: (i) Let $n\geq k\geq 5$ and let
$t=\left\lfloor\frac{k-1}{2}\right\rfloor$. Let $G$ be a $2$-connected
$n$-vertex $\mathcal{C}_{\geq k}$-free graph with the maximum $\mathcal{P}(G)$
where $\mathcal{P}(G)$ is feasible. Then, $G\in
\mathcal{G}^1_{n,k}=\{W_{n,k,s}=K_{s}\vee ((n-k+s)K_1\cup K_{k-2s}): 2\leq
s\leq t\}$. (ii) Let $n\geq k\geq 4$ and let
$t=\left\lfloor\frac{k}{2}\right\rfloor-1$. Let $G$ be a connected $n$-vertex
$P_{k}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is
feasible. Then, $G\in \mathcal{G}^2_{n,k}=\{W_{n,k-1,s}=K_{s}\vee
((n-k+s+1)K_1\cup K_{k-2s-1}): 1\leq s\leq t\}.$ (iii) Let $G$ be a connected
$n$-vertex $M_{k+1}$-free graph with the maximum $\mathcal{P}(G)$ where
$\mathcal{P}(G)$ is feasible. Then, $G\cong K_n$ when $n=2k+1$ and $G\in
\mathcal{G}^3_{n,k}=\{K_s\vee ((n-2k+s-1)K_1\cup K_{2k-2s+1}):1\leq s\leq k\}$
when $n\geq 2k+2$. Directly derived from these three main results, we obtain a
series of applications in Tur\'an-type problems, generalized Tur\'an-type
problems, powers of graph degrees in extremal graph theory, and problems
related to spectral radius, and signless Laplacian spectral radius in spectral
graph theory.

Combinatorics