Let G be a simple graph with vertical bar V(G)vertical bar = n and no isolated vertices. Let a he its stability number. We study invariants of the edge-ring of G that can he interpreted as invariants of G. If G has a cover by maximum stable sets we are able to prove the inequality a <= n/2. As a byproduct we prove that if G is vertex-critical, then at alpha <= (n - vertical bar A vertical bar)/2, where A is the intersection of all the minimum vertex covers of G. We estimate the smallest number of vertices in any maximal stable set of G to obtain a bound for the depth of the edge-ring of G.
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