# This is a joint work done by Omar A. AbuGhneim from Jordan University (Email: o_abughneim@yahoo.com) and Ken W. Smith from # Sam Houston State University (Email: kenwsmith@shsu.edu). # Regular (96,19,2,4) partial difference sets provide 12 nonisomorphic Strongly regular (96,19,2,4) Graphs. # The set of regular (96,19,2,4) partial difference sets are listed in the file "Regular (96,19,2,4) Partial Difference Sets". # Each one of these regular (96,19,2,4) partial difference sets provides a strongly regular (96,19,2,4) graph. # For instance: the second (96,19,2,4) partial difference set in group [ 96, 70 ], the second (96,19,2,4) partial difference set # in group [ 96, 195 ], and the fifth (96,19,2,4) partial difference set in group [ 96, 227 ] provide the same strongly regular # (96,19,2,4) graph (provide three strongly regular (96,19,2,4) graphs but they are isomorphic and we consider them the same). # In here when we said the second (96,19,2,4) partial difference set in group [ 96, 70 ] we mean the second # (96,19,2,4) partial difference set in group [ 96, 70 ] in the list of (96,19,2,4) partial difference sets in the file # "Regular (96,19,2,4) Partial Difference Sets" and we denote this partial difference set by [ [ 96, 70 ], 2 ]. # We use this notation for all other partial difference sets. # To construct these strongly regular (96,19,2,4) graphs and to check which are isomorphic, we used the GRAPE package for GAP, # Version 4.6.1, 2012 by L.H. Soicher, http://www.maths.qmul.ac.uk/~leonard/grape/ . # We give the program that uses the regular (96,19,2,4) partial difference sets to construct strongly regular # (96,19,2,4) graphs and to check if these strongly regular (96,19,2,4) graphs are isomorphic. # The inequivalent Regular (96,19,2,4) Partial Difference Sets in group [96, 64] are # B1:=[ 2, 3, 4, 7, 9, 13, 17, 32, 37, 47, 50, 66, 67, 70, 74, 75, 79, 84, 89 ]; # B2:= [ 2, 3, 6, 7, 8, 9, 13, 23, 32, 36, 39, 55, 57, 59, 64, 71, 72, 86, 94 ]; # The following program construct the corresponding graphs and check if they are isomorphic. ## LoadPackage( "grape" ); ## g:=SmallGroup(96,64); ## e:=Elements(g); ## B1:= [ 2, 3, 4, 7, 9, 13, 17, 32, 37, 47, 50, 66, 67, 70, 74, 75, 79, 84, 89 ]; ## B2:= [ 2, 3, 6, 7, 8, 9, 13, 23, 32, 36, 39, 55, 57, 59, 64, 71, 72, 86, 94 ]; ## D1:=[]; ## D2:=[]; ## for j in B1 ## Add(D1,e[j]); ## od; ## for j in B2 ## Add(D2,e[j]); ## od; ## G1:=CayleyGraph(g, D1 ); ## G2:=CayleyGraph(g, D2 ); ## IsIsomorphicGraph(G1,G2) # The 12 nonisomorphic Strongly Regular (96,19,2,4) Graphs are: # Strongly Regular (96,19,2,4) Graph number 1. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference set: # [ [ 96, 64 ], 1 ] # Strongly Regular (96,19,2,4) Graph number 2. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference sets: # [ [ 96, 64 ], 2 ] # [ [ 96, 70 ], 1 ] # [ [ 96, 71 ], 1 ] # [ [ 96, 190 ], 1 ] # [ [ 96, 195 ], 1 ] # [ [ 96, 195 ], 4 ] # [ [ 96, 195 ], 8 ] # [ [ 96, 227 ], 1 ] # [ [ 96, 227 ], 3 ] # Strongly Regular (96,19,2,4) Graph number 3. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference sets: # [ [ 96, 70 ], 2 ] # [ [ 96, 195 ], 2 ] # [ [ 96, 227 ], 5 ] # Strongly Regular (96,19,2,4) Graph number 4. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference sets: # [ [ 96, 186 ], 1 ] # [ [ 96, 190 ], 2 ] # [ [ 96, 195 ], 3 ] # [ [ 96, 197 ], 1 ] # [ [ 96, 226 ], 1 ] # Strongly Regular (96,19,2,4) Graph number 5. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference sets: # [ [ 96, 186 ], 2 ] # [ [ 96, 195 ], 12 ] # [ [ 96, 197 ], 2 ] # [ [ 96, 226 ], 5 ] # Strongly Regular (96,19,2,4) Graph number 6. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference sets: # [ [ 96, 195 ], 5 ] # [ [ 96, 195 ], 10 ] # [ [ 96, 227 ], 2 ] # Strongly Regular (96,19,2,4) Graph number 7. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference sets: # [ [ 96, 195 ], 6 ] # [ [ 96, 195 ], 7 ] # [ [ 96, 227 ], 4 ] # Strongly Regular (96,19,2,4) Graph number 8. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference set: # [ [ 96, 195 ], 9 ] # Strongly Regular (96,19,2,4) Graph number 9. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference set: # [ [ 96, 195 ], 11 ] # Strongly Regular (96,19,2,4) Graph number 10. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference set: # [ [ 96, 226 ], 2 ] # Strongly Regular (96,19,2,4) Graph number 11. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference set: # [ [ 96, 226 ], 3 ] # Strongly Regular (96,19,2,4) Graph number 12. # This Strongly Regular (96,19,2,4) Graph rises from the following Regular (96,19,2,4) partial difference set: # [ [ 96, 226 ], 4 ]