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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.

# Level 3

If $$G$$ is a graph with 61 vertices and 60 edges, then:

Find the $$10 \times 10$$ adjacency matrix $$A$$ of the graph above.

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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