The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Given a graph H and a positive integer n, the extremal number, ex(n, H), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. A keyring \(C_s(k)\) is a \((k+s)\)-edge graph obtained from a cycle of length k by appending s leaves to one of its vertices. This paper determines the extremal number and finds the extremal graphs for the blow-ups of keyrings \(C_s(k)\) (\(k\ge 3\), \(s\ge 1\)) when n is sufficiently large. For special cases when \(k=0\) or \(s=0\), the extremal number of the blow-ups of the graph \(C_s(0)\) (a star) has been determined by Erdös et al. (J Comb Theory Ser B 64:89–100, 1995) and Chen et al. (J Comb Theory Ser B 89: 159–171, 2003), while the extremal number and extremal graphs for the blow-ups of the graph \(C_0(k)\) (a cycle) when n is sufficiently large has been determined by Liu (Electron J Combin 20: P65, 2013).

图H的爆炸是通过用相同大小的集团替换H中的每个边缘而获得的图，其中集团的新顶点都不同。给定一个图H和一个正整数n，极值ex（n，  H）是n个不包含H作为子图的顶点在图上的最大边数。密钥环\（C_s（k）\）是\（（k + s）\）-边图，是通过将s附加到长度为k的循环而获得的离开其顶点之一。当n足够大时，本文确定了极值数量并找到了密匙环\（C_s（k）\）（\（k \ ge 3 \），\（s \ ge 1 \））爆炸的极值图。对于\（k = 0 \）或\（s = 0 \）的特殊情况，Erdös等人已经确定了图\（C_s（0）\）（星形）爆炸的极值数目。（J Comb Theory Ser B 64：89-100，1995）和Chen等。（J Comb Theory Ser B 89：159–171，2003），而当n为n时，图\（C_0（k）\）（一个周期）爆炸的极值数和极值图。 Liu已确定其足够大（Electron J Combin 20：P65，2013）。