Abstract
Colin de Vediere introduced an interesting linear algebraic invariant μ(G) of graphs. He proved that μ(G)≤2 if and only ifG is outerplanar, and μ(G)≤3 if and only ifG is planar. We prove that if the complement of a graphG onn nodes is outerplanar, then μ(G)≥n−4, and if it is planar, then μ(G)≥n−5. We give a full characterization of maximal planar graphs whose complementsG have μ(G)=n−5. In the opposite direction we show that ifG does not have “twin” nodes, then μ(G)≥n−3 implies that the complement ofG is outerplanar, and μ(G)≥n−4 implies that the complement ofG is planar.
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