I'm certainly not saying linear algebra is useless with planar graphs. It really depends on what is meant by graph theory vs. linear algebra here. If GT means

I'm certainly not saying linear algebra is useless with planar graphs. It really depends on what is meant by graph theory vs. linear algebra here. If GT means we can fix an embedding to the space, then we have additional information such as number of crossings. However, it seems like a bit arbitrary restriction.