In this paper the sharpness of an upper bound, due to Merris, on the independence number of a graph is investigated. Graphs that attain this bound are called Merris graphs. Some families of Merris graphs are found, including Kneser graphs K(v, 2) and non-singular regular bipartite graphs. For example, the Petersen graph and the Clebsch graph turn out to be Merris graphs. Some sufficient conditions for non-Merrisness are studied in the paper. In particular it is shown that the only Merris graphs

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... among the joins are the stars. It is also proved that every graph is isomorphic to an induced subgraph of a Merris graph and conjectured that almost all graphs are not Merris graphs.