Let $G$ be a simple graph. The clique complex, $\Delta(G)$ of the graph $G$ is the simplicial complex given by the collection of all complete subgraph of $G.$ Now we define the graph homology $H^{Gr}_\ast(G) : = H_\ast(\Delta(G))$ (the simplicial homology of $\Delta(G)$).
Next, define a contractible graph as follows:
A family $\mathcal{F}$ of graphs $G_1, G_2, \cdots, G_n ,\cdots$ is called contractible if
(1) The trivial graph, $\ast \in \mathcal{F}.$
(2) Any graph of $\mathcal{F}$ can be obtained from the trivial graph by finite series of contractible transformations $\{T_1, T_2, T_3, T_4\}$
where $T_1$: deleting of vertex $v$. A vertex $v$ of a graph $G$ can be deleted, if $N_G(v):= \{ u \in V(G): \text{ the edge }[uv] \in E(G)\} \in \mathcal{F}.$
$T_2:$ Gluing of a vertex $v$. If a subgraph $G_1$ of a graph $G$ is contractible, $G_1 \in \mathcal{F}$ the the vertex $v$ can be glued to the graph $G$ in such a manner that $N_G(v) =G_1,$
$T_3:$ deleting of an edge $[v_1v_2]$. The edge $[v_1v_2]$ of $G$ can be deleted if $N_G(v_1)\cap N_G(v_2)\in \mathcal{F}.$
$T_4:$ Gluing of an edge $[v_1v_2]$. Let two vertices $v_1$ and $v_2$ of a graph $G$ be non-adjacent. The edge $[v_1v_2]$ of $G$ can be glued if $N_G(v_1)\cap N_G(v_2)\in \mathcal{F}.$
Any graph $G \in \mathcal{F}$ is called a contractible graph.
$\mathbf{Question:}$ Let $G$ be a graph such that $H^{Gr}_\ast(G) =0 $ for $\ast>0$ and $\mathbb{Z}$ for $\ast=0.$ Then can we prove that the suspension graph $S(G)$ (suspension is considered as the topological unreduced suspension on the vertices of $G$) is contractible in above sense.
Thank you so much in advance. Any help will be appreciated.
