How can I create an adjacency matrix from a long list of source/target pairs?

You can prepare the data for the adjacency matrix like this:

# create the "class-tuples" by
# joining the dataframe with itself
df_cross= df.merge(df, on='student', suffixes=['_left', '_right'])
# remove the duplicate tuples
# --> this will get you a upper / or lower
# triangular matrix with diagonal = 0
# if you rather want to have a full matrix
# just change the >= to == below
del_indexer= (df_cross['class_left']>=df_cross['class_right'])
df_cross.drop(df_cross[del_indexer].index, inplace=True)
# create the counts / lists
grouby_obj= df_cross.groupby(['class_left', 'class_right'])
result= grouby_obj.count()
result.columns= ['value']
# if you want to have lists of student names
# that have the course-combination in
# common, you can do it with the following line
# otherwise just remove it (I guess with a 
# dataset of the size you mentioned, it will
# consume a lot of memory)
result['students']= grouby_obj.agg(list)

The full output looks like this:

Out[133]: 
                        value                     students
class_left class_right                                    
Chemistry  English          1                 [John Smith]
           Math             2  [John Smith, Jenny Simmons]
           Physics          1              [Jenny Simmons]
English    Math             1                 [John Smith]
           Physics          1                [Sarah Blume]
Math       Physics          1              [Jenny Simmons]

You then can use @piRSquared's method to pivot it, or do it like this:

result['value'].unstack()

Out[137]: 
class_right  English  Math  Physics
class_left                         
Chemistry        1.0   2.0      1.0
English          NaN   1.0      1.0
Math             NaN   NaN      1.0

Or, if you also want the names:

result.unstack()
Out[138]: 
              value                   students                                              
class_right English Math Physics       English                         Math          Physics
class_left                                                                                  
Chemistry       1.0  2.0     1.0  [John Smith]  [John Smith, Jenny Simmons]  [Jenny Simmons]
English         NaN  1.0     1.0           NaN                 [John Smith]    [Sarah Blume]
Math            NaN  NaN     1.0           NaN                          NaN  [Jenny Simmons]