Laplacian Eigenvectors of Graphs pp 29-47 | Cite as

# Eigenfunctions and Nodal Domains

In the previous chapter we have seen that (due to the Perron-Frobenius Theorem) the eigenfunctions of the first eigenvalue *λ*1 have all entries positive (or negative) for a generalized Laplacian matrix M of a connected graph *G*. Fiedler [67] has shown that for eigenfunctions of the smallest nonzero eigenvalue of a graph the subgraph induced by nonpositive vertices (i.e., vertices with nonpositive function values) and the subgraph induced by nonnegative vertices are both connected. In other words, an eigenfunction of the second eigenvalue has exactly two *weak nodal domains* (also called *weak sign graphs*).

## Keywords

Connected Graph Sign Pattern Boundary Edge Characteristic Edge Sign Graph## Preview

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