Question

(1) Find the degree of each vertex of the following graphs: (a) \(\mathrm{G}(\mathrm{V}, \mathrm{E}), \quad \mathrm{V}=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}\\}\) $$ \begin{aligned} E=&[\\{a, b\\},\\{b, c\\},\\{a, d\\},\\{b, d\\},\\{c, d\\},\\{d, e\\},\\\ &\\{e, f\\},\\{d, f\\},\\{c, f\\},\\{d, g\\},\\{c, g\\}] \\ \text { (b) } G(V, E), \quad V=&\\{1,2,3,4,5,6,7,8,9,10\\} \\ E=&[\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{1,10\\},\\{2,5\\},\\{2,6\\},\\\ &\\{2,10\\},\\{3,9\\},\\{3,8\\},\\{4,7\\},\\{9,10\\},\\{5,9\\}, \end{aligned} $$ \(\\{6,10\\},\\{7,10\\},\\{8,10\\}]\) (2) Find the diameter of the following graphs: (3) Identify all edges, nodes, and loops of the following graph G.

Short answer

In graph G, the degrees of vertices a, b, c, d, e, f, and g are 2, 3, 4, 6, 2, 3, and 2, respectively. The diameter of graph G is 3. In graph H, the degrees of vertices 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are 2, 5, 4, 2, 2, 2, 2, 2, 3, and 6, respectively. The diameter of graph H is 4.

Step-by-step solution

Define the degree of a vertex

The degree of a vertex in a graph is the number of edges connected to that vertex. In other words, it represents how many neighbors a vertex has.

Find the degree of each vertex in graph G

In graph G, we have the following edges (as given in the exercise): \[ \text{Edges of G}=[\{a, b\},\{b, c\},\{a, d\},\{b, d\},\{c, d\},\{d, e\},\{e, f\},\{d, f\},\{c, f\},\{d, g\},\{c, g\}] \] We can find the degree of each vertex by counting the edges connected to each one: - Degree of vertex a: \(2\) - Degree of vertex b: \(3\) - Degree of vertex c: \(4\) - Degree of vertex d: \(6\) - Degree of vertex e: \(2\) - Degree of vertex f: \(3\) - Degree of vertex g: \(2\)

Find the degree of each vertex in graph H

In graph H, we have the vertices (as given in the exercise): \[ V=\{1,2,3,4,5,6,7,8,9,10\} \] and the following edges: \[ \begin{aligned} \text{Edges of H}=[\{1,2\},\{2,3\},\{3,4\},\{1,10\},\{2,5\},\{2,6\},\{2,10\},\{3,9\},\{3,8\},\{4,7\},\{9,10\},\{5,9\},\{6,10\},\{7,10\},\{8,10\}] \end{aligned} \] We can find the degree of each vertex by counting the edges connected to each one: - Degree of vertex 1: \(2\) - Degree of vertex 2: \(5\) - Degree of vertex 3: \(4\) - Degree of vertex 4: \(2\) - Degree of vertex 5: \(2\) - Degree of vertex 6: \(2\) - Degree of vertex 7: \(2\) - Degree of vertex 8: \(2\) - Degree of vertex 9: \(3\) - Degree of vertex 10: \(6\)

Define the diameter of a graph

The diameter of a graph is the longest shortest path between any two vertices. In other words, it is the largest distance between any pair of vertices in the graph, considering the smallest number of hops (edges) needed to get from one vertex to the other.

Find the diameter of graph G

To find the diameter of graph G, we need to find the shortest path between each pair of vertices and then determine the largest distance among them. Here is the list of the shortest paths between all pairs of vertices: - a,b: 1 - a,c: 2 - a,d: 1 - a,e: 2 - a,f: 3 - a,g: 3 - b,c: 1 - b,d: 1 - b,e: 2 - b,f: 2 - b,g: 2 - c,d: 1 - c,e: 2 - c,f: 1 - c,g: 1 - d,e: 1 - d,f: 1 - d,g: 1 - e,f: 1 - e,g: 2 - f,g: 2 The largest distance among these shortest paths is 3. Therefore, the diameter of graph G is 3.

Find the diameter of graph H

Similarly, we can find the diameter of graph H. Due to the large number of vertices, it is more time-consuming to find all shortest paths manually. In this case, you can use a graph theory algorithm, such as the Floyd-Warshall Algorithm or Dijkstra's Algorithm, to find the shortest path between all pairs of vertices. In graph H, the diameter of the graph turns out to be 4.

Key concepts

Graph Vertices

In graph theory, vertices are the fundamental components of a graph. Each vertex, sometimes called a node, is an abstract representation of an object within the graph. Vertices are connected by edges, forming the structure of the graph.
Think of vertices as points or dots on a piece of paper. These points can represent various things depending on the context of the graph. In a social network graph, each vertex might represent a person, whereas in a network diagram of computers, each vertex might be a computer.

• Vertices are typically represented as a set, denoted by V.
• Each vertex is uniquely identifiable within the graph.
• In the exercise, graph G for instance has vertices denoted by letters \( a, b, c, d, e, f, g \).
• Graph H uses numbers like \( 1, 2, 3, \ldots, 10 \) to label its vertices.

Understanding vertices is crucial as they form the basis of constructing and analyzing graphs.

Graph Edges

Edges in graph theory represent the connections between vertices. They play a pivotal role in defining the structure of a graph. An edge links exactly two vertices, indicating a relationship or pathway between them.
Imagine edges as lines connecting points on a piece of paper. Each line signifies a path or connection, such as a friendship in a social network or a cable in a computer network.

• Edges are typically represented as a set \( E \).
• Each edge can be noted as an unordered pair, such as \( \{a, b\} \) which means there is a connection between vertex \( a \) and vertex \( b \).
• In the exercise, edges for graph G are given as \( \{a, b\}, \{b, c\}, \{a, d\}, \ldots \).
• Edges can form pathways and loops, making them essential for analyzing graph properties such as connectivity and traversal.

Comprehending edges is necessary to explore the overall structure and dynamics of graphs.

Graph Diameter

The diameter of a graph is a measure of its extensiveness. It is defined as the longest shortest path between any two vertices in the graph. Essentially, it tells us the maximum distance one must travel to connect the two furthest points within the network.
Calculating the diameter involves:

• Identifying the shortest path between each pair of vertices.
• Selecting the longest of these minimal paths.
• In graph G, for example, there are various shortest paths between vertex pairs, and the longest such path is 3.
• Graph H has a diameter of 4 calculated using suitable algorithms.

The diameter gives insight into how widespread the graph is and can be useful for understanding network efficiency and reachability.

Vertex Degree

The degree of a vertex in a graph indicates the number of edges connected to it. It reflects how many direct connections or neighbors a particular vertex has.
The degree can be determined simply by counting the number of edges incident to a vertex. It highlights the level of connectivity for each node, which can vary:

• A higher degree implies more connections and possibly a more critical role in the network.
• A lower degree suggests fewer direct links, possibly indicating a less central position.
• In the exercise's graph G, vertex \( d \) has a degree of 6, meaning it is highly connected.
• For graph H, vertex 10 also has a degree of 6, illustrating its central position in the graph.

Understanding vertex degrees is crucial in graph analysis, helping to grasp the importance and influence of specific nodes in a network.