Tree Identification and Minimal Edges in Connected Graphs
Consider a connected graph where the number of vertices is \(n\) and the number of edges is \(n-1\). This property is a characteristic feature of trees. Answer the following: (a) Identify why a connected graph with 8 vertices and 7 edges must be acyclic. Describe the characteristics confirming its acyclic nature. (b) Prove that in […]
Identifying Tree Structures in Graph Theory
A tree is defined as an acyclic connected graph. Explain why a tree with \(n\) vertices must always have exactly \(n-1\) edges.
Modeling Transportation Networks Using Graph Theory
Consider a city transportation network represented as a directed weighted graph. The network must meet specific regulatory and operational constraints for safety and efficiency.\n\n(a) Formulate the constraints that ensure no intersection (node) has more than 4 incoming roads and at least 2 outgoing roads, while also ensuring the network does not contain cycles that may […]