8 people total. Seated around a circular table. Hibachi dinner? You can only talk to the 2 people seated directly next to you.
Each person has a connection value with each other person. eg : #1 and #3 are friendly and have a "conversation value" of 7. #1 and #2 are not friends, and have a "conversation value" of 3.
Picture a star diagram to represent all connections between all people. How do you represent this data? Matrix? How do you arrange the table to maximize the seating arrangement to result in the highest total connection value?
How do I actually solve an example scenario with actual sample values? How do I store this data? How do I solve the maximization?
If I represented all connections between all people I don't get a "star diagram" but a complete graph, and the conversation values make this a weighted complete graph. This can be represented as a symmetric matrix of the values as you propose.
The problem of maximising the conversation value sum around the table is essentially (not just equivalent to) the travelling salesman problem – which is usually formulated as a minimisation problem, but we can negate the conversation values in this case.