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# Graphs

Whether you're finding the shortest path between two locations or modeling a social network, graphs are are a critical tool for storing data and exploring connections.

If \(G\) is a graph with 61 vertices and 60 edges, then:

Input \( \det \left( \frac{( A^2 + A - K )}{2} \right) \) as your answer.

**Details and assumptions**

- \(K\) is the \(10 \times 10\) matrix which has \(1\) for all entries.

Which of the following adjacency matrices represent a bipartite graph?

\[ \text{A} = \left(\begin{array}{ccccc}0 & \color{red}{1} & \color{red}{1} & 0 & 0 \\ \color{red}{1} & 0 & 0 & \color{red}{1} & 0 \\\color{red}{1} & 0 & 0 & 0 & 0 \\0 & \color{red}{1} & 0 & 0 & \color{red}{1} \\0 & 0 & 0 & \color{red}{1} & 0\end{array}\right), \quad \text{B} = \left(\begin{array}{ccccc}0 & \color{blue}{1} & 0 & \color{blue}{1}& 0 \\ \color{blue}{1} & 0 & \color{blue}{1} & 0 & \color{blue}{1} \\0 & \color{blue}{1} & 0 & \color{blue}{1} & 0 \\ \color{blue}{1} & 0 & \color{blue}{1} & 0 & \color{blue}{1} \\0 & \color{blue}{1} & 0 & \color{blue}{1} & 0\end{array}\right) \]

\[ \text{C} = \left(\begin{array}{ccccc}0 & 0 & \color{green}{1} & 0 & 0 \\0 & 0 & \color{green}{1} & 0 & 0 \\\color{green}{1} & \color{green}{1} & 0 & \color{green}{1} & \color{green}{1} \\0 & 0 & \color{green}{1} & 0 & 0 \\0 & 0 & \color{green}{1} & 0 & 0\end{array}\right)\]

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