The number of connected sparsely edged graphs. II. Smooth graphs and blocks

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Abstract
A smooth graph is a connected graph without endpoints; f(n, q) is the number of connected graphs, v(n, q) is the number of smooth graphs, and u(n, q) is the number of blocks on n labeled points and q edges: Wk, Vk, and Uk are the exponential generating functions of f(n, n + k), v(n, n + k), and u(n, n + k), respectively. For any k ⩾ 1, our reduction method shows that Vk can be deduced at once from Wk, which was found for successive k by the computer method described in our previous paper. Again the reduction method shows that Uk must be a sum of powers (mostly negative) of 1 ‐ X and, given this information, we develop a recurrence method well suited to calculate Uk for successive k. Exact formulas for v(n, n + k) and u(n, n + k) for general n follow at once.

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