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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 graphsdoi:10.13001/1081-3810.1108 fatcat:22hkamnxqnfstdpvsvarswufdu