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Every graph can be embedded in the three dimensional Euclidean space, but not necessarily in the plane. This gives rise to the concept of planar graphs, one of the classical objects in topological graph theory. In this talk, we will discuss planar graphs and its analogs, for instance, linkless embeddable graphs. Colin de Verdiere introduced a graph parameter based on the ideas in spectral geometry of Riemannian surfaces. This parameter turns out to characterize graphs with these topological properties of graphs. We will summarize these ideas and report some recent progress on this parameter.