Are almost all graphs determined by their spectrum? ★★★


\begin{problem} Are almost all graphs uniquely determined by the spectrum of their adjacency matrix? \end{problem}

Keywords: cospectral; graph invariant; spectrum

The sum of the two largest eigenvalues ★★

Author(s): Gernert

\begin{problem} Let $G$ be a graph on $n$ vertices and let $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$ be the eigenvalues of $G$. Is $\lambda_1 + \lambda_2 \le n$? \end{problem}

Keywords: eigenvalues; spectrum

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