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Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the adjacency operator are contained in the interval $$[-2\sqrt{q}, 2\sqrt{q}].$$ The reason for this ...

Let $G$ be a digraph such that there is an unique directed walk of length $k$ between any two vertices. Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $...