Question

Assume that all the weights on the edges of a graph are positive. Prove that if no two edges in a graph have the same weight, then the MCST is unique. (Hint: Assume \(G\) has more than one minimal spanning tree, Order the edges, and consider the smallest subscript \(k\) of an edge that belongs to some, but not all, minimal spanning trees.)

Short answer

If no two edges have the same weight, the MCST is unique due to ordered edge selection.

Step-by-step solution

Understand the Problem

We need to prove that if no two edges in a graph have the same weight, the Minimum Cost Spanning Tree (MCST) of that graph is unique. This means there is only one way to connect all the vertices of the graph (using the edges) such that the total weight is minimized.

Consider Assumptions

Assume for contradiction that there are two different MCSTs for a given graph. Let's denote these two trees as T1 and T2. Both T1 and T2 have the same total weight but differ in their edge sets.

Order the Edges by Weight

List all edges of the graph in increasing order of their weights, as no two edges have the same weight. This categorically orders them uniquely by their weights.

Find the Smallest Index Edge

Identify the smallest index (lightest) edge, say e_k, that is present in one tree, say T1, but not in T2. This is possible because we've assumed two different sets of edges exist for T1 and T2.

Analyze the Implication

Since e_k is in T1 but not in T2, T2 must have another edge that connects the same vertices as e_k to avoid forming a cycle and maintain connectivity. However, e_k was chosen because it's the smallest such edge, hence any replacement in T2 must have a larger weight.

Conclude the Contradiction

The presence of a heavier replacement edge in T2 would result in a total weight that's greater than T1, contradicting our assumption that T1 and T2 have the same total weight. This contradiction implies our assumption that more than one MCST exists is false under these conditions.

Key concepts

Graph Theory

Graph theory is a fascinating area of mathematics focused on studying graphs, which are structures made up of vertices (or nodes) connected by edges. It provides tools to model relationships through these graphs, applicable in various fields such as computer science, biology, and social networks.
In the context of a Minimum Cost Spanning Tree (MCST), graph theory helps us understand how to connect all the vertices in a graph using the minimum possible total edge weight. A spanning tree is a subgraph that includes all the vertices of the original graph but only enough edges to maintain connectivity without forming cycles.
When applying graph theory to MCST problems, one often utilizes algorithms like Kruskal's or Prim's, which systematically select edges to form a spanning tree with the least weight efficiently.

Edge Weight Uniqueness

Edge weight uniqueness plays a crucial role in the uniqueness of the Minimum Cost Spanning Tree. Here, it means that no two edges in the graph share the same weight. This uniqueness ensures a strict order of edge selection based on increasing weights.
When each edge has a distinct weight, sorting the edges for MCST algorithms becomes straightforward. This unique ordering is pivotal because it prevents ambiguity during edge selection.
Consider a scenario where all edge weights are distinct. As we construct the spanning tree, any deviation in edge selection affects the total weight, providing no leeway for alternate trees with equal weights. This property directly enforces the uniqueness of the MCST.

Spanning Tree Uniqueness

The concept of spanning tree uniqueness in a graph is closely tied to edge weight uniqueness. When no two edges have the same weight, it implies that any minimum spanning tree formed with the least possible weight is unique.
To prove spanning tree uniqueness, assume there are two different MCSTs with the same total weight. Upon careful inspection, due to distinct edge weights, one can identify an edge in one tree missing in the other. Replacing this edge would always result in a heavier total weight, thus violating our assumption.
This leads us to conclude that given each edge has a unique weight, only one configuration of edges will yield the minimum total weight, ensuring the spanning tree's uniqueness.

MCST Proof

Proving the uniqueness of a Minimum Cost Spanning Tree in a graph with distinct edge weights involves strategic reasoning and a contradiction approach. Here's how it unfolds:
First, assume there's more than one minimal spanning tree (MST) for a graph, despite all edges having distinct weights. Let's call these two MSTs T1 and T2, both with identical total weights but differing in certain edges.
Upon scrutinizing these edge differences, identify the smallest indexed edge present in T1 but absent in T2. Since no two edges have the same weight, adding this edge to T2 will increase its total weight beyond T1's, contradicting the condition that both have equal weight.
This contradiction demonstrates that it's impossible to have multiple MCSTs under unique edge weights, thus proving the MCST's uniqueness decisively.