On the Eigenvalues of k-Degenerate Graphs

초록

A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k. The well-known Brualdi-Solheid problem asks for the maximum spectral radius of a graph belonging to a specified class of graphs and the characterization of extremal graphs. In this paper, we first characterize k-degenerate graphs with the minimum least eigenvalue. As an application, we give an answer to the Brualdi-Solheid problem for k-degenerate bipartite graphs. In addition, we give a new proof to determine the maximum signless Laplacian spectral radius and characterize the corresponding extremal graphs among k-degenerate graphs. We also raise a problem related to the second largest eigenvalues of k-degenerate graphs and solve this problem for 1-degenerate graphs.

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