The King (still) needs your help

An ambitious king plans to build five new cities in the wilderness, connected by a network of roads, so that any city can be reached from any other. He expects the annual tax revenues from each city to be numerically equal to the square of the population. Road maintenance will be expensive, though; the annual cost for each road is expected to be numerically equal to the product of the populations of the two cities that road connects. The project is considered viable as long as the tax revenues exceed the cost, regardless of the populations of the various cities (as long as at least one city has any inhabitants at all). How many options does the king have, that is, how many graphs (up to isomorphism) can he use to get a viable layout, with the vertices representing the cities and the edges representing the roads?

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