Book cover for Discrete Mathematics with Applications

Discrete Mathematics with Applications

Thomas Koshy

ISBN #124211801

1st Edition

3,631 Questions

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Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explores trees in graph theory, emphasizing that every tree with n vertices has n - 1 edges, and examines various specific cases such as complete graphs, complete bipartite graphs, and regular trees. It also introduces spanning tree algorithms, especially DFS, to extract tree structures from connected graphs. Applications include molecular bonding in hydrocarbons and solving the n-queens puzzle. Understanding these concepts is critical for both theoretical and algorithmic aspects of graph theory.

Learning Objectives

1

Explain and verify fundamental properties of trees, including that a tree with n vertices has n - 1 edges.

2

Analyze various types of graphs (complete, bipartite, and regular) to determine when they form trees.

3

Demonstrate how spanning tree algorithms (especially DFS-based) work for connected graphs.

4

Apply tree concepts to real-world problems such as determining the bonding in hydrocarbon molecules and solving the n-queens puzzle.

Key Concepts

CONCEPT

DEFINITION

Tree

A connected graph with no cycles, where any two vertices are connected by exactly one simple path.

Edge-Vertex Relationship

For any tree containing n vertices, the number of edges is exactly n - 1.

Complete Graph (Kn)

A graph in which every pair of distinct vertices is connected by a unique edge.

Complete Bipartite Graph (Km,n)

A graph whose vertices can be divided into two disjoint sets such that every vertex in one set is connected to every vertex in the other set.

Regular Graph

A graph where every vertex has the same degree. An r-regular graph has every vertex of degree r.

Spanning Tree

A subgraph that is a tree including all the vertices of the original connected graph.

Eccentricity

The length of the longest simple path from a given vertex to any other vertex in the tree.

Center of a Tree

The vertex (or vertices) with the minimum eccentricity within the tree.

DFS (Depth-First Search)

An algorithm for traversing or searching through graph or tree data structures by exploring as far as possible along each branch before backtracking.

Example Problems

Example 1

Determine if each graph is a tree. IMAGE IS NOT AVAILABLE TO COPY

Example 2

Determine if each graph is a tree. IMAGE IS NOT AVAILABLE TO COPY

Example 3

Determine if each graph is a tree. IMAGE IS NOT AVAILABLE TO COPY

Example 4

Determine if each graph is a tree. IMAGE IS NOT AVAILABLE TO COPY

Example 5

Determine if each graph is a tree. IMAGE IS NOT AVAILABLE TO COPY
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Step-by-Step Explanations

QUESTION

Verify that a tree with n vertices has n - 1 edges.

STEP-BY-STEP ANSWER:

Step 1: Begin with the base case. A single vertex (n = 1) has no edges, which satisfies the formula 0 = 1 - 1.
Step 2: Assume a tree with n vertices has n - 1 edges.
Step 3: When adding a new vertex to an existing tree, it must be connected by exactly one edge (to maintain the acyclic property). This increases the vertex count by 1 and the edge count by 1.
Step 4: Therefore, the new tree has (n + 1) vertices and (n - 1) + 1 = n edges, which is consistent with the formula.
Final Answer: Every tree with n vertices has exactly n - 1 edges.

Tree Edge Count Verification

QUESTION

How many bonds does the hydrocarbon molecule CnH2n+2 have, assuming each carbon atom has a degree of four?

STEP-BY-STEP ANSWER:

Step 1: Recognize that in an alkane (CnH2n+2), each carbon atom forms a total of 4 bonds (either with other carbons or hydrogens).
Step 2: Let the number of carbon-carbon bonds be represented by x. Since the carbon skeleton forms a tree, by the tree property, x = n - 1.
Step 3: The remaining bonds from carbon are with hydrogen atoms. Each carbon contributes (4 - its carbon-carbon bonds), so total carbon bonds equal 4n.
Step 4: Taking into account that each carbon-carbon bond is counted twice when summing degrees (once at each carbon), we have: 4n = 2*(carbon–carbon bonds) + (carbon–hydrogen bonds).
Step 5: Substitute x for the carbon–carbon bonds and note that the molecular formula provides 2n+2 hydrogen atoms (i.e., carbon–hydrogen bonds). Thus, 4n = 2(n - 1) + (2n + 2).
Step 6: Simplify: 4n = 2n - 2 + 2n + 2, which confirms 4n = 4n.
Step 7: Finally, the total number of bonds is the sum of the carbon–carbon bonds (n - 1) and the carbon–hydrogen bonds (2n + 2).
Final Answer: The molecule has (n - 1) + (2n + 2) = 3n + 1 bonds.

Hydrocarbon Molecule Bonds

QUESTION

For what values of n is the complete graph Kn a tree?

STEP-BY-STEP ANSWER:

Step 1: Recall that a complete graph Kn has n(n - 1)/2 edges.
Step 2: For Kn to be a tree, it must have exactly n - 1 edges.
Step 3: Set up the equality: n(n - 1)/2 = n - 1.
Step 4: For n > 1, divide both sides by (n - 1) to get n/2 = 1, which implies n = 2.
Step 5: Also note that for n = 1, K1 trivially forms a tree.
Final Answer: Kn is a tree only when n = 1 or n = 2.

Complete Graph as a Tree

QUESTION

Outline the basic steps of constructing a spanning tree using depth-first search in a connected graph.

STEP-BY-STEP ANSWER:

Step 1: Start at an arbitrary vertex and mark it as visited.
Step 2: Explore its adjacent vertices in a specified order (e.g., alphabetical order).
Step 3: For each adjacent vertex that has not yet been visited, add the edge to the spanning tree and recursively perform DFS from that vertex.
Step 4: If a vertex has no unvisited adjacent vertices, backtrack to the previous vertex and continue exploring.
Step 5: Repeat the process until all vertices have been visited and added to the spanning tree.
Final Answer: The DFS-based algorithm systematically explores all vertices, building a spanning tree that connects all vertices without forming cycles.

Spanning Tree Construction via DFS

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Common Mistakes

  • Assuming that any connected graph with n vertices automatically qualifies as a tree without checking for cycles.
  • Forgetting that each added vertex in a tree must be connected by exactly one edge, which is crucial in the inductive proof of e = n - 1.
  • Overlooking the double counting involved in determining the number of bonds in a hydrocarbon molecule when using the degree of carbon atoms.
  • Misinterpreting the DFS spanning tree algorithm by not appropriately backtracking when all adjacent vertices have been visited.