Some Geometric Aspects of Graphs and their Eigenfunctions
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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.