a feeling about the value of Parke-Taylors and star graph in the analyzing a more complicated Cayley function. For next-to-Hamiltonian graph, |T(C\^[nH]{}\_n)|= [Cat]{}\_[n-3]{}+[Cat]{}\_[n-3]{} +\_[r=0]{}\^[n-5]{} |T(C\^[nH]{}\_[4+r]{})| [Cat]{}\_[n-5-r]{}, where we have used $|T(C^{H}_n)|={\rm Cat}_{n-2}$. So we obtain |T(C\^[nH]{}\_n)|=. The counting of the number of poles is simple. Using the additional edge, we have $(n-3)\times 2^1-1$ more poles. Then |P(C\^[nH]{}\_n)|= +1+(n-3)2\^1-1 =-4. Similarly, |T(C\^[nS]{}\_[n]{})| = ,|P(C\^[nS]{}\_[n]{})| =32\^[n-4]{}-1.

More complicated Cayley functions can be analyzed with the help of simpler ones. Different kinds of Cayley functions have different pole structures $P(C_n)$ and Feynman diagram structures $T(C_n)$. For $n\geq 6$, there are too many poles or Feynman diagrams for any Cayley function to put them here. We just make a list showing the number of poles and Feynman diagrams below.

  $n$                    $|T(C_n)|$                               $|P(C_n)|$
  ----- --------------------------------------------- ----------------------------------
  4                           2                                       2
  5                          5,6                                     5,6
  6                       14,18,24                                 9,11,14
  7                  42,56,60,76,84,120                       14,17,18,21,23,30
  8      132,180,200,222,248,280,288,324,408,480,720   20,24,26,28,29,32,33,36,41,47,62
  9                                                   

  : \[tab:widgets\] $|T(C_n)|$ and $|P(C_n)|$ for $n\leq 9$

The lists are always like $|T(C_n^H)|,|T(C_n^{nH})|,\cdots,|T(C_{n}^{nS})|,|T(C_n^S)|$( $|P(C_n^H)|,|P(C_n^{nH})|,\cdots,|P(C_{n}^{nS})|,|P(C_n^S)|$ ). Note that $|T(C_n)|,|P(C_n)|$ are only rough descriptions of Cayley functions. Two Cayley functions from two kinds may share the same $|P(C_n)|$ or $|T(C_n)|$ , seen in figure \[samenumber\].

A more accurate way is to map a Cayley 