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Chapter 11: Problem 86

Let \(G\) be a weighted connected graph in which all edge weights are different. Prove that there is exactly one spanning tree of minimum weight.

Short Answer

Expert verified
The reason why there's only one minimum weight spanning tree when all edge weights are different is due to the uniqueness of each weight. If there were two different minimum spanning trees, then this would create a contradiction where we could create a spanning tree of even smaller weight, which opposes the definition of a minimum weight spanning tree.

Step by step solution

01

Understand the definitions

A graph \(G\) is a set of vertices and edges. A connected graph is a graph where every pair of vertices is connected. A weighted graph is a graph where each edge has a weight associated. A spanning tree of \(G\) is a subgraph that includes all the vertices of \(G\) and forms a tree. A minimum weight spanning tree is the spanning tree with the least total edge weight.
02

Assume contradictorily

Assume that there are at least two minimum weight spanning trees, say \(T\) and \(T'\).
03

Show contradiction

There must be an edge \(e\) in \( T\) that is not in \(T'\). Remove \(e\) from \(T\), creating two disjoint trees. Then, there must exist an edge \(e'\) in \(T'\) that reconnects these two trees without creating a cycle (because \(T'\) is a tree). But, since the weights of all edges are different, the weight of \(e\) cannot be equal to \(e'\), and without loss of generality, let's assume \(w(e) < w(e')\). By replacing \(e\) with \(e'\) in \(T\), we have created a spanning tree with weight less than \(T\), contradicting the assumption that \(T\) was a minimum weight spanning tree.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

spanning tree
In the realm of graph theory, a spanning tree serves as a critical concept. Consider a connected, undirected graph. A spanning tree of this graph is a subgraph that includes all the vertices connected by just enough edges to form a tree structure.
A tree is essentially a connected graph with no cycles. Hence, a spanning tree has exactly \(n-1\) edges where \(n\) stands for the number of vertices. This minimal number of edges ensures a connected graph without any loops.
  • Spanning trees are crucial for optimizing network designs since they lay out the most efficient pathways.
  • An essential characteristic is that removing any edge from a spanning tree would disconnect the graph.
Understanding spanning trees helps in grasping more complex concepts like minimum spanning trees, which further optimize network costs.
weighted graph
Weighted graphs enhance the study of graphs by assigning a value called 'weight' to each edge. These weights can represent various aspects such as distance, cost, or time, depending on the context.
In a weighted graph, each edge \(e\) is associated with a numerical value \(w(e)\). These weights allow for the calculation of different paths and their cumulative weights, which are crucial for problems like finding the shortest path or the minimum spanning tree.
  • The weights play a critical role in defining the efficiency of a path within a graph.
  • Applications of weighted graphs include networking, where weights could represent bandwidth or delay.
The concept of weighted graphs forms the basis for solving many real-world optimization problems.
minimum spanning tree
A minimum spanning tree (MST) is an extension of a spanning tree where the total weight of the edges is minimized. This extends the concept of a spanning tree by focusing not just on simplicity but also on cost-effectiveness.
An MST of a graph ensures that all vertices are connected with the least possible total edge weight.
  • It is unique in a graph with distinct edge weights, meaning there is only one possible MST configuration in such cases.
  • Finding an MST is vital for applications that require the least cost, like electrical circuit design or laying cables.
By understanding MSTs, we can effectively solve problems where minimizing costs or resources is necessary, while still maintaining full connectivity.
connected graph
A connected graph is a foundational element in graph theory. It is defined as a graph where there exists a path between every pair of vertices, irrespective of how they connect.
This property implies that the graph forms a single unified structure without any isolated parts. For a graph to have a spanning tree, it must be connected.
  • Even one missing edge could split the graph into two, transforming it from connected to disconnected.
  • Connected graphs ensure comprehensive traversal, which is crucial for network connectivity and optimization tasks.
The study of connected graphs underpins the exploration of more complex structures like spanning and minimum spanning trees, providing a baseline for analyzing network efficiency.

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Most popular questions from this chapter

Let \(V=\\{1,2, \ldots, 20\\}\) be the set of the first 20 positive integers. Consider the graphs whose vertex set is \(V\) and whose edge sets are defined below. For earth graph, investigate whether the graph (i) is connected (if not connected, determin" the connected components), (ii) is bipartite, (iii) has an Eulerian trail, and (iv) has a Hamilton path. (a) \(\\{a, b\\}\) is an edge if and only if \(a+b\) is even. (b) \(\\{a, b\\}\) is an edge if and only if \(a+b\) is odd. (c) \(\\{a, b\\}\) is an edge if and only if \(a \times b\) is even. (d) \(\\{a, b\\}\) is an edge if and only if \(a \times b\) is odd. (e) \(\\{a, b\\}\) is an edge if and only if \(a \times b\) is a perfect square. (f) \(\\{a, b\\}\) is an edge if and only if \(a-b\) is divisible by 3 .

Let \(x\) and \(y\) be vertices of a general graph, and suppose that there is a closed trail containing both \(x\) and \(y\). Must there be a cycle containing both \(x\) and \(y\) ?

Let \(G\) be a graph with degree sequence \(\left(d_{1}, d_{2}, \ldots, d_{n}\right) .\) Prove that, for each \(k\) with \(0

Let \(G\) be a connected graph of order 6 with degree sequence \((2,2,2,2,2,2)\). (a) Determine all the nonisomorphic induced subgraphs of \(G\). (b) Determine all the nonisomorphic spanning subgraphs of \(G\). (b) Determine all the nonisomorphic subgraphs of order 6 of \(G\).

Let \(G\) be a graph that has a Hamilton path which joins two vertices \(u\) and \(v\). Is the Hamilton path a DFS-tree rooted at \(u\) for \(G ?\) Could there be other DFS-trees?
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