There appears to be no standard term for a simple connected graph with exactly n edges, though the words "polynema" (Kyrmse) and "polyedge" (Muñiz 2011) have been proposed. The numbers of n-polynemas for n = 2, 3 ... are 1, 1, 3, 5, 12, 30, 79, 227, ... (OEIS A002905). An n-polynema has n + 1 - γ nodes, where γ is its circuit rank. Polynemas are related to a graphical construction problem called the match problem.
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