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In this paper, we introduce new attributes of low-density parity-check graphs, degree-set and degree-sum, which explains the connectivity of a check node with more detail. Then, we consider more specified ensembles defined by degree-set/sum distribution of the graph than one defined by the degree distribution. As the degree set/sum distribution varies, tradeoff between the performance at the waterfall and that at the error-floor is observed. The trade-off point can be managed by controlling degree-set/sum distribution.