DELETE rows which are not referenced in other table
Quoting the manual:


  There are two ways to delete rows in a table using information
  contained in other tables in the database: using sub-selects, or
  specifying additional tables in the USING clause. Which technique is
  more appropriate depends on the specific circumstances.


Bold emphasis mine. Using information that is not contained in another table is a tad bit tricky, but there are easy solutions. From the arsenal of standard techniques to ...


Select rows which are not present in other table


... a NOT EXISTS anti-semi-join is probably simplest and most efficient for DELETE:

DELETE FROM link_group lg
WHERE  NOT EXISTS (
   SELECT FROM link_reply lr
   WHERE  lr.which_group = lg.link_group_id
   );


Assuming (since table definitions are not provided) link_group_id as column name for the primary key of link_group.

The technique @Mihai commented works as well (applied correctly):

DELETE FROM link_group lg
USING  link_group      lg1
LEFT   JOIN link_reply lr ON lr.which_group = lg1.link_group_id
WHERE  lg1.link_group_id = lg.link_group_id
AND    lr.which_group IS NULL;


But since the table expression in the USING clause is joined to the target table (lg in the example) with a CROSS JOIN, you need another instance of the same table as stepping stone (lg1 in the example) for the LEFT JOIN, which is less elegant and typically slower.