The Tur\'{ a}n number of a graph $H$, $\text{ ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{ F}$ of graphs, the suspension $\mathcal{ F}+v$ of $\mathcal{ F}$ is the graph obtained by connecting the vertex $v$ to all vertices of $F$ for each $F\in \mathcal{ F}$. For two integers $k\ge1$ and $r\ge2$, let $H_i$ be a graph containing a critical edge with chromatic number $r$ for any $i\in\{ 1,\ldots,k\}$, and let $H=\{ H_1,\ldots,H_k\}+v$. In this paper, we determine $\text{ ex}(n, H)$ and characterize all the extremal graphs for sufficiently large $n$. This generalizes a result of Chen, Gould, Pfender and Wei on intersecting cliques. We also obtain a stability theorem for $H$, extending a result of Roberts and Scott on graphs containing a critical edge.
Categories: math.CO