The problem describes a network of 5 fast food restaurants: Haile's Hell Hamburgers (H), Mattingley's Macaroons (M), Taylor's Tacos (T), Palmer's Parmas (P), and Arnott's Biscuits (A). The network diagram represents the paths that directly connect these restaurants. We are asked to: a) Construct a communication matrix $C$ to represent the above diagram. b) Explain what $C^2$ represents in this context.

1. Problem Description

The problem describes a network of 5 fast food restaurants: Haile's Hell Hamburgers (H), Mattingley's Macaroons (M), Taylor's Tacos (T), Palmer's Parmas (P), and Arnott's Biscuits (A). The network diagram represents the paths that directly connect these restaurants. We are asked to:

a) Construct a communication matrix $C$ to represent the above diagram.

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

2. Solution Steps

a) Constructing the Communication Matrix $C$:

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 order of the restaurants is A, H, M, P, T.

- A: A is connected to H and M. So, $C_{12} = 1$ and $C_{13} = 1$. All other entries in the first row are

0. - H: H is connected to A, M, and T. So, $C_{21} = 1$, $C_{23} = 1$, and $C_{25} = 1$. All other entries in the second row are

0. - M: M is connected to A, H, and P. So, $C_{31} = 1$, $C_{32} = 1$, and $C_{34} = 1$. All other entries in the third row are

0. - P: P is connected to M. So, $C_{43} = 1$. All other entries in the fourth row are

0. - T: T is connected to H. So, $C_{52} = 1$. All other entries in the fifth row are

0.

Therefore, 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) What does $C^2$ represent?

$C^2 = C \times C$. The entry $(C^2)_{ij}$ represents the number of paths of length 2 from restaurant $i$ to restaurant $j$. In other words, it tells you how many ways you can get from restaurant $i$ to restaurant $j$ by going through exactly one other restaurant.

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 travel from restaurant $i$ to restaurant $j$ using exactly one intermediate restaurant.