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Let H be a fixed graph on n vertices. Let f_H(G) = 1 iff the input graph G on n vertices contains H as a (not necessarily induced) subgraph. Let α_H denote the cardinality of a maximum independent set of H. In this paper we show: Q(f_H) = Ω(√(α_H · n)), where Q(f_H) denotes the quantum query complexity of f_H. As a consequence we obtain a lower bounds for Q(f_H) in terms of several other parameters of H such as the average degree, minimum vertex cover, chromatic number, and the criticalarXiv:1509.06361v2 fatcat:eydgqkabuvhbhdxdxrwarfnqcu