  The stability number alpha(G) of a graph G is the cardinality of a maximum
stable set in G, xi(G) denotes the size of core(G), where core(G) is the
intersection of all maximum stable sets of G. In this paper we prove that for a
graph G without isolated vertices, the following assertions are true: (i) if
xi(G)< 2, then G is quasi-regularizable; (ii) if G is of order n and alpha(G) >
(n+k-1)/2, for some k > 0, then xi(G) > k, and xi(G) > k+1, whenever n+k-1 is
even. The last finding is a strengthening of a result of Hammer, Hansen, and
Simeone, which states that alpha(G) > n/2 implies xi(G) > 0. G is a
Koenig-Egervary graph if n equals the sum of its stability number and the
cardinality of a maximum matching. For Koenig-Egervary graphs, we prove that
alpha(G) > n/2 holds if and only if xi(G) is greater than the size of the
neighborhood of core(G). Moreover, for bipartite graphs without isolated
vertices, alpha(G) > n/2 is equivalent to xi(G) > 1. We also show that Hall's
marriage Theorem is valid for Koenig-Egervary graphs, and it is sufficient to
check Hall's condition only for one specific stable set, namely, for core(G).
