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Given a Graph $G = (V=[n],E)$, if a symmetric matrix $B$ fits $G$, it has non-zero diagonal elements and 0 on off-diagonal entries if $\{i,j\}$ are non-edge in $G$. Let \begin{equation} R(G) = \min ...

Let $\mathbf{x}_1$ and $\mathbf{x}_2$ be column vectors with entries belonging to the set $\{0,1\}$. In addition, let $\mathbf{A}$ be the adjacency matrix of an arbitrary graph. I want to find vectors ...

Let $\Sigma=(G,\sigma)$ be an unbalanced signed graph with the underlying connected graph $G=(V,E)$ and $\sigma:E\rightarrow \{-1,1\}$, the signing function. Let the Laplacian eigenvalues of $\Sigma$ ...