For star graph, all particles except the center one and $n$ are end points, which are symmetric. Each of them corresponds to a line from center point to it. Any nontrivial subset of these lines must make up a connected subgraph which corresponds to a pole. So there are $2^{n-2}-2$ poles in star graph. What’s more, in star graph, any two connected subgraphs are compatible if and only if one is contained in the other. So start from two-particle pole, the next subgraph is compatible to the former if and only if it contains the former. So there are $(n-2)!$ sets of $n-3$ compatible poles, any of which corresponds to a multi-peripheral Feynman diagram . The exact pole sets and Feynman diagram sets are given by \[p of S\] P(C\^[[S]{}]{}\_n(1))&=&{s\_[1,i\_1,,i\_m]{}|[i\_1,,i\_m]{}{2,,n-1},1mn-3 }, T(C\^[[S]{}]{}\_n(1))&=&{ | 2,3,,n-1 } .

Starting from $n=6$, Cayley functions of new kind comes out, see . We can also extend it to general $n$, next-to-Hamiltonian graph seen in figure \[nH graph\] and next-to-Star graph seen in figure \[nS graph\].

With more patience, we can also list out their pole sets and Feynman diagram sets. Here we just show a nontrivial use of the recursion about $T(C_n)$ and get 