Suppose we have two graphs $\Gamma_1,\Gamma_2$ which are isomporhic to two pentagons and with vertices $v_1,\dots,v_5$ and $v_1',\dots,v_5'$ respectively. Let us construct a new graph $\Gamma$ with vertex set $V(\Gamma_1)\cup V(\Gamma_2)$ and edge set equal to: $$E(\Gamma_1)\cup E(\Gamma_2)\cup E(\lbrace v_1,v_2\rbrace, \Gamma_2)\cup E(\lbrace v_1',v_2'\rbrace, \Gamma_1)$$ Here $E(\lbrace v_1,v_2\rbrace, \Gamma_2)$ means that we consider an edge $\lbrace u,v\rbrace$ for each $v\in \lbrace v_1,v_2\rbrace$ and each $u\in V(\Gamma_2)$. Hence, our new graph $\Gamma$ is constructed via gluing our two pentagons in some ''fancy way'' via adding edges with respect to two vertices of each pentagon.
Consider the flag complex of $\Gamma$, i.e. the simplicial complex obtained after adding an $(n-1)$-cell for each complete graph of $\Gamma$ with $n$ vertices. I want to compute the connectivity of $\Gamma$.
I know that both $\Gamma_1$ and $\Gamma_2$ are homotopic to $\mathbb{S}^1$ so they are $0$-connected, i.e. they are path-connected. However, I have no idea on how to study $\Gamma$. I expect to increase the connectivity somehow, since joins of simplicial complexes increse the connectivity and this construction looks like a ''partial join'' where we have not added all the edges between the two initial complexes.
Thanks for your help.