Let $A$ be the adjacency matrix of a directed graph with $n$ vertices and spectral radius $\lambda$. Let $I$ be the $n \times n$ identity matrix and let $e \in \mathbb{R}^n$ be the vector of 1's. For $\gamma<\lambda$, the matrix $I-\gamma A$ is invertible and the Bonacich centrality vector is defined as $$ B(\gamma) = (I-\gamma A)^{-1}e.$$ Is it possible to find another matrix $A$ of 0's and 1's satisfying the above? If so, can $e$ be replaced by another vector in $\mathbb{R}^n$ such that it is no longer possible to find an alternative $A$?
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