A necessary and sufficient eigenvector condition for a connected graph to be bipartite.
The spectral radius and the maximum degree of irregular graphs.
Bounds on the Turán density of PG(3, 2).
More counterexamples to the Alon-Saks-Seymour and rank-coloring conjectures.
Principal eigenvectors of irregular graphs.
Connectivity, toughness, spanning trees of bounded degree, and the spectrum of regular graphs
The eigenvalues of graphs are related to many of its combinatorial properties. In his fundamental work, Fiedler showed the close connections between the Laplacian eigenvalues and eigenvectors of a graph and its vertex-connectivity and edge-connectivity. We present some new results describing the connections between the spectrum of a regular graph and other combinatorial parameters such as its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.
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