Limit points of eigenvalues of (di)graphs

Zhang, Fuji ; Chen, Zhibo

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 3, pp. 895-902.

Résumé

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove: 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph $D$, the set of limit points of eigenvalues of iterated subdivision digraphs of $D$ is the unit circle in the complex plane if and only if $D$ has a directed cycle. 3. Every limit point of eigenvalues of a set $\mathcal{D}$ of digraphs (graphs) is a limit point of eigenvalues of a set $\ddot{\mathcal{D}}$ of bipartite digraphs (graphs), where $\ddot{\mathcal{D}}$ consists of the double covers of the members in $\mathcal{D}$. 4. Every limit point of eigenvalues of a set $\mathcal{D}$ of digraphs is a limit point of eigenvalues of line digraphs of the digraphs in $\mathcal{D}$. 5. If $M$ is a limit point of the largest eigenvalues of graphs, then $-M$ is a limit point of the smallest eigenvalues of graphs.

Détail

Classification : 05C50, 15A48
Mots-clés : limit point; eigenvalue of digraph (graph); double cover; subdivision digraph; line digraph