\chapter{Results}\label{chap4}
{ 
\section{Complete Graphs}\label{sec:resultsKn}
{
The Star Analysis process represents a total departure from the exhaustive search method. Beginning with Harris and Harris and progressing up to this point, each improvement built upon the work that had gone before without changing the underlying algorithm, apart from optimizing the search space. With Star Analysis, we started from the ground up, so our first concern was assuring that the process worked correctly. Therefore, we began our Star Analysis test runs on the complete K$_N$ graphs generated previously by the processes described in Chapter~\ref{chap2}. Recall that graphs up to $n = 10$ were generated by the parallel processing method with radical region restriction, the most optimized system prior to Star Analysis. By repeating the process using the Star Analysis module, we were able to confirm that the same number of both total graphs and isomorphic families were being generated. We also of course confirmed that the same crossing numbers were obtained (Table ?). 

 \begin{table}[h]
	\centering
		\begin{tabular}{|c|cccccc|} \hline
	$n$ 		         &  5 & 6 &  7 &  8 &    9 &    10   \\ \hline
	Graphs generated & 12 & 4 & 76 & 20 & 4,560 & 56,618  \\ \hline
	Iso families     &  1 & 1 &  5 & 3  & 1,453 &  5,679  \\ \hline
		\end{tabular}
	\caption{Isomorphic Families of Minimal K$_{n}$}
	\label{tbl:isofamilies}
\end{table}
   
The only discrepency occurred at K$_{10}$; the same number of isomorphic families were found, but there were slightly fewer total graphs with Star Analysis than with Parallel/Radical. This discrepency remains unresolved, but we are reviewing the graphs generated by Star Analysis to be sure that no graphs were inappropriately deleted. This issue does not affect the progression to K$_{11}$, since all isomorphic families are available to provide feeder graphs, but could potentially be a problem with larger $n$. 

Table ? shows the results of these test runs as compared with the results from parallel processing/radical region restricton. Time approximations for Star Analysis are based on the entire process, including all the scripts and the isomorphic testing by NAUTY.

 \begin{table}[h]
	\centering
		\begin{tabular}{|c|cccccc|} \hline
	$n$ 		         &  5 & 6 &  7 &  8 &    9 &    10   \\ \hline
	Graphs generated & 12 & 4 & 76 & 20 & 4,560 & 56,618  \\ \hline
	Iso families     &  1 & 1 &  5 & 3  & 1,453 &  5,679  \\ \hline
		\end{tabular}
	\caption{Isomorphic Families of Minimal K$_{n}$}
	\label{tbl:isofamilies}
\end{table}   

 For the larger values of $n$, NAUTY became the rate-limiting step in the process. We were able to generate the Path Lists and obtain the value for minimal crossing number relatively quickly. Table ? shows the approximate times for Path List generation and crossing number calculation alone. NAUTY is reputed to be the fastest iso-testing program available, but improvements in processing time here would be most welcome. 

 \begin{table}[h]
	\centering
		\begin{tabular}{|c|cccccc|} \hline
	$n$ 		         &  5 & 6 &  7 &  8 &    9 &    10   \\ \hline
	Graphs generated & 12 & 4 & 76 & 20 & 4,560 & 56,618  \\ \hline
	Iso families     &  1 & 1 &  5 & 3  & 1,453 &  5,679  \\ \hline
		\end{tabular}
	\caption{Isomorphic Families of Minimal K$_{n}$}
	\label{tbl:isofamilies}
\end{table}   

)%\end section {Complete Graphs}


 
 
      
  	        
 
 
 