The problem describes a network of 5 fast food restaurants and the direct paths connecting them. We are asked to: a) Construct a communication matrix $C$ representing the network. b) Explain what $C^2$ represents in this context. c) Determine the minimum number of steps required to reach every restaurant from any given starting restaurant.

1. Problem Description

The problem describes a network of 5 fast food restaurants and the direct paths connecting them. We are asked to:

a) Construct a communication matrix $C$ representing the network.

b) Explain what $C^2$ represents in this context.

c) Determine the minimum number of steps required to reach every restaurant from any given starting restaurant.

2. Solution Steps

a) The communication matrix $C$ is a 5x5 matrix where $C_{ij} = 1$ if there is a direct path from restaurant $i$ to restaurant $j$, and $C_{ij} = 0$ otherwise. The restaurants are in the order A, H, M, P, T.

Looking at the diagram:

- A is connected to H and M.

- H is connected to A, M, and T.

- M is connected to A, H, and P.

- P is connected to M.

- T is connected to H.

Thus, the communication matrix $C$ is:

$C = \begin{bmatrix}

0 & 1 & 1 & 0 & 0 \\

1 & 0 & 1 & 0 & 1 \\

1 & 1 & 0 & 1 & 0 \\

0 & 0 & 1 & 0 & 0 \\

0 & 1 & 0 & 0 & 0

\end{bmatrix}$

b) The matrix $C^2 = C \times C$. The element $(C^2)_{ij}$ represents the number of paths of length 2 from restaurant $i$ to restaurant $j$. In other words, it indicates the number of ways to get from restaurant $i$ to restaurant $j$ by passing through exactly one intermediate restaurant.

c) To determine the minimum number of steps required to ensure someone can get to every restaurant, we need to find the smallest $n$ such that $C + C^2 + ... + C^n$ has no zero entries. This means that there is a path of length at most $n$ between any two restaurants.

We can analyze the network visually. Starting from T, we must go to H. Then from H, we can go to A, M, or T. From P, we can only go to M.

- A is connected to H, M

- H is connected to A, M, T

- M is connected to A, H, P

- P is connected to M

- T is connected to H

From T, the furthest restaurant to reach is P. To get from T to P, one could go T -> H -> M -> P. So the number of steps is

3. From P, the furthest restaurant to reach is T. To get from P to T, one could go P -> M -> H -> T. So the number of steps is

3.

However, starting from the matrix:

$C = \begin{bmatrix}

0 & 1 & 1 & 0 & 0 \\

1 & 0 & 1 & 0 & 1 \\

1 & 1 & 0 & 1 & 0 \\

0 & 0 & 1 & 0 & 0 \\

0 & 1 & 0 & 0 & 0

\end{bmatrix}$

$C^2 = \begin{bmatrix}

2 & 1 & 1 & 1 & 0 \\

1 & 3 & 1 & 0 & 0 \\

1 & 1 & 3 & 0 & 1 \\

1 & 0 & 0 & 1 & 0 \\

1 & 0 & 1 & 0 & 1

\end{bmatrix}$

$C + C^2 = \begin{bmatrix}

2 & 2 & 2 & 1 & 0 \\

2 & 3 & 2 & 0 & 1 \\

2 & 2 & 3 & 1 & 1 \\

1 & 0 & 1 & 1 & 0 \\

1 & 1 & 1 & 0 & 1

\end{bmatrix}$

Now, let us compute $C^3$.

$C^3 = C^2 \times C = \begin{bmatrix}

2 & 1 & 1 & 1 & 0 \\

1 & 3 & 1 & 0 & 0 \\

1 & 1 & 3 & 0 & 1 \\

1 & 0 & 0 & 1 & 0 \\

1 & 0 & 1 & 0 & 1

\end{bmatrix} \begin{bmatrix}

0 & 1 & 1 & 0 & 0 \\

1 & 0 & 1 & 0 & 1 \\

1 & 1 & 0 & 1 & 0 \\

0 & 0 & 1 & 0 & 0 \\

0 & 1 & 0 & 0 & 0

\end{bmatrix} = \begin{bmatrix}

2 & 3 & 3 & 1 & 1 \\

4 & 1 & 4 & 1 & 3 \\

1 & 3 & 2 & 3 & 1 \\

1 & 0 & 1 & 0 & 1 \\

1 & 2 & 2 & 0 & 0

\end{bmatrix}$

So after 3 steps, it is not possible to reach all the destinations as the entry $(4,2)$ is still

0. So $n>3$. The minimum number of steps is

3.

3. Final Answer

a) $C = \begin{bmatrix}

0 & 1 & 1 & 0 & 0 \\

1 & 0 & 1 & 0 & 1 \\

1 & 1 & 0 & 1 & 0 \\

0 & 0 & 1 & 0 & 0 \\

0 & 1 & 0 & 0 & 0

\end{bmatrix}$

b) $C^2$ represents the number of paths of length 2 between any two restaurants. $(C^2)_{ij}$ indicates the number of ways to get from restaurant $i$ to restaurant $j$ by passing through exactly one intermediate restaurant.

c) The minimum number of required steps to ensure someone can get to every restaurant is

3.