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We analyze two essential problems arising from edge-based graph partitioning. We show that one of them is an NP-hard problem but the other is in P, presenting a novel methodology that links linear algebra theory to the graph problems as a part of proving the facts. This is a significant trial in that linear algebra, which has been mostly adopted as a theoretical analysis tool, is practically applied to solving actual graph problems. As a result of the linear algebraic manipulation, we could devise a linear-time algorithm for the problem in P.