is getting the adjacency matrix costly?

Is it costly to repeatedly access the adjacency matrix with
graph_tool.spectral.adjacency() ? i.e. does it get computed every time I
make a call to it? If so, is it really costly to do?

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Is it costly to repeatedly access the adjacency matrix with
graph_tool.spectral.adjacency() ? i.e. does it get computed every time
I make a call to it?

Yes.

If so, is it really costly to do?

It takes time O(E), where E is the number of edges.

Best,
Tiago