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Recently, Ma, Qian and Shi determined the maximum size of an n-vertex graph with given fractional matching number s and maximum degree at most d. Motivated by this result, we determine the maximum number of -cliques in a graph with given fractional matching number and minimum degree, which generalizes Shi and Ma's result about the maximum size of a graph with given fractional matching number and minimum degree at least one. We also determine the maximum number of complete bipartite graphs in a graph with prescribed fractional matching number and minimum degree.
Li et al. (Wed,) studied this question.