Beautiful approach. I got the 3 connections at each corner vertex, and 6 in the middle vertex, 1 for every other vertex, too. Then I went on to reason that 19 consists of 10 uneven and 9 even numbers, since 22 is even, and only three even numbers or two uneven and one even yield an even result, I went on to find a configuration on the grid, satisfying this. Which proved to be a tedious effort. However, convinced that this ought to be true, I didn't conceive of the summing up, which as presented by you, proves very elegant.

Thanks for such a logical explanation and method, these magic riddles often have a habit of getting me in a bad mood, for their lack of solving methodology.