Some Geometric Aspects of Graphs and their Eigenfunctions
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  • Abstract:

    In this paper we study graph eigenfunctions and eigenvalues with respect to two mathematical theories: nodal regions and fiber products. First we show how the nodal region theory for the Laplacian carries over to graphs, and give some applications. Second we study the affect of taking fiber products of graphs from the point of view of their second eigenvalue; we make some numerical calculations which show that fiber products of certain arithmetic graphs can have smaller second eigenvalues than the original class of graphs, and that sometimes twisting the fiber product (in non-arithmetic ways) produces graphs with smaller second eigenvalues.