fig-example-graph.tex

\begin{figure}
    \begin{subfigure}[b]{0.5\linewidth}
        \centering
        \begin{tikzpicture}[-stealth,align=center,every node/.style={circle, draw}]
            \sf % use sans-serif font
            \node[]              (V1) {1};
            \node[, right=of V1] (V2) {2};
            \node[, below=of V1] (V3) {3};
            \node[, right=of V3] (V4) {4};
            \node[, right=of V4] (V5) {5};
            \draw (V1) to [bend left] (V2);
            \draw (V1) to [bend left] (V3);
            \draw (V2) to [bend left] (V4);
            \draw (V2) to [bend left] (V5);
            \draw (V3) to [bend left] (V1);
            \draw (V3) to [bend left] (V4);
            \draw (V4) to [bend left] (V5);
        \end{tikzpicture}
        \caption{Graph.}
    \end{subfigure}
    \begin{subfigure}[b]{0.4\linewidth}
        \centering
        \setlength{\arraycolsep}{2pt}
        \renewcommand{\arraystretch}{0.8}
        $$
            \begin{bNiceMatrix}[first-row,first-col]
              & 1     & 2     & 3     & 4     & 5     \\
            1 &       & \grbT & \grbT &       &       \\
            2 &       &       &       & \grbT & \grbT \\
            3 & \grbT &       &       & \grbT &       \\
            4 &       &       &       &       & \grbT \\
            5 &       &       &       &       &       \\
            \end{bNiceMatrix}
        $$
        \caption{Adjacency matrix.}
    \end{subfigure}

    \caption{Example directed graph and its adjacency matrix.}
    \label{fig:example-graph}
\end{figure}


\begin{align*}
    \grbv{s} =
    \begin{bNiceMatrix}[first-row]
        1 & 2     & 3     & 4     & 5     \\
        \grbT &   &       &       &       \\
    \end{bNiceMatrix}
    \qquad
    \grbv{f}                                                                                & =
    \begin{bNiceMatrix}[first-row]
        1 & 2     & 3     & 4     & 5     \\
        & \grbT & \grbT &       &       \\
    \end{bNiceMatrix}
    \\
    %
    \grbv{f} = \grbv{f} \grbanypair \grbm{A}                           \qquad \grbv{f}      & =
    \begin{bNiceMatrix}[first-row]
        1 & 2     & 3     & 4     & 5     \\
        \grbT &       &       & \grbT & \grbT \\
    \end{bNiceMatrix}
    \\
    %
    \grbv{f}\grbmask{\grbneg \grbv{s}} = \grbv{f} \grbanypair \grbm{A} \qquad \grbv{f}      & =
    \begin{bNiceMatrix}[first-row]
        1 & 2     & 3     & 4     & 5     \\
        &       &       & \grbT & \grbT \\
    \end{bNiceMatrix}
    \\
    %
    \grbv{f}\grbmask{\grbneg \grbv{s}} = \grbv{f} \grbminfirstjone \grbm{A} \qquad \grbv{f} & =
    \begin{bNiceMatrix}[first-row]
        1 & 2     & 3     & 4     & 5     \\
        &       &       & 2 & 2 \\
    \end{bNiceMatrix}
    \\
    %
    \grbv{f}\grbmask{\grbneg \grbv{s}} = \grbv{f} \grbanyfirstjone \grbm{A} \qquad \grbv{f} & =
    \begin{bNiceMatrix}[first-row]
        1 & 2     & 3     & 4     & 5     \\
        &       &       & 2 & 2 \text{ or } 4 \\ % benign race condition
    \end{bNiceMatrix}
\end{align*}