Maximising the number of independent sets in connected graphs

The Turán connected graph TCn,α is obtained from α cliques of size ⌊nα⌋ or ⌈nα⌉ by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph TCn,α was shown recently by Bruyère and Mélot to maximise the number of independent sets among connected graphs of order n and independence number α. We prove a generalisation of this result by showing that TCn,α in fact maximises the number of independent sets of any fixed cardinality β≤α. Several results (both old and new) on the number of independent sets or maximum independent sets follow as corollaries