How to get the adjacency matrix of the dual of $G$ without pen and paper?

Solution 1:

The dual of a planar graph depends on how it is embedded in the plane: "The same planar graph can have non-isomorphic dual graphs." (See Wikipedia illustration.)

Nonisomorphic duals of graph embedded two ways

Although this graph is not regular, it can be modified to obtain a cubic (3-regular) graph with more than one planar embedding and associated nonisomorphic dual graphs. Replace the left- and right-hand vertices of degree two in the top diagram with a "diamond" subgraph, connecting to its top and bottom nodes to produce all nodes of degree 3:

A diamond replacing degree 2 with degree 3

Now the outer face of the top graph will have its maximum degree 10, but the corresponding replacement in the alternative embedding yields a maximum degree 7 (shared by the outer face and one inner face). So the dual (multi-)graph depends on the embedding and cannot be inferred only from the adjacency matrix of the original graph.

So I'd say in the general sense you've asked, it cannot be done without pencil-and-paper or some other visualization of the embedding.

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