Question

Construct connected graphs of the following sorts: (a) All graphs of five vertices with at least seven edges (b) All cubic graphs with at most eight vertices (c) One 4 -regular graphs of six vertices (d) Three 5 -regular graphs of eight vertices

Short answer

Construct graphs as per the conditions: (a) at least 7 edges with 5 vertices, (b) cubic with up to 8 vertices, (c) one 4-regular with 6 vertices, (d) three 5-regular with 8 vertices.

Step-by-step solution

Understanding Graphs and Their Components

A **graph** is a collection of vertices (or nodes) and edges connecting pairs of these vertices. A graph is **connected** if there is a path between any pair of vertices. In a **regular graph**, every vertex has the same degree (number of edges). Let's construct the graphs accordingly.

Construct Graphs of Five Vertices with at Least Seven Edges

A graph with five vertices can have a maximum of ten edges. Construct graphs starting from any four connected vertices forming a cycle and then adding more edges. Consider the following graphs: 1. Add one extra edge to the cycle to form a pentagon plus one diagonal (6 edges), then add another edge to form a pentagon with two diagonals (7 edges). 2. Connect all vertices, forming a complete graph with ten edges, and remove any three edges to ensure it's connected with at least 7 edges.

Construct All Cubic Graphs with at Most Eight Vertices

A cubic graph is 3-regular, meaning each vertex has a degree of 3. Start with fewer vertices and add edges while maintaining 3-degree for each vertex. 1. For 4 vertices: the complete graph with four vertices (K4) is cubic. 2. For 6 vertices: consider the 1-skeleton of the octahedron. 3. For 8 vertices: the cube can be used, where each vertex connects to three vertices.

Construct One 4-Regular Graph of Six Vertices

A 4-regular graph with 6 vertices will have each vertex connected by 4 edges. The total number of edges should be 12 (6 vertices x 4 degrees / 2, due to double counting each edge). One example is the Petersen graph, which has 10 vertices and 15 edges, but we seek a smaller example. Create a new graph from scratch with trial and error or modify an existing smaller graph to match these conditions.

Construct Three 5-Regular Graphs of Eight Vertices

A 5-regular graph on 8 vertices would have each vertex connected to five others, totaling 20 edges. Consider existing graphs or combinations such as the complete bipartite graph mechanisms to arrive at such structures. Possible graphs include semi-structured combinations, expanded bicliques maintaining the 5-regularity.

Key concepts

Connected Graph

A connected graph is one of the fundamental concepts in graph theory. It offers insight into how well-integrated the components of a network are. In simple terms, a graph is connected if, for any two vertices in the graph, there exists at least one path that links them. This means that you can start at any vertex and, through a series of edges, reach any other vertex in the graph.

Understanding connected graphs is crucial because they guarantee that there is no isolated vertex or subgroup. This concept is significant in applications like transportation networks, where each location (vertex) must be accessible from others. Moreover, ensuring a graph remains connected while manipulating edges offers challenges and opportunities in designing robust systems.

• All vertices must be reachable from any other via a single path.
• A connected graph does not have any disconnected vertices or subgraphs.
• Examining the connectivity of a system helps in identifying potential points of failure or improvements.

Regular Graph

A regular graph is a special type of graph where each vertex has the same number of edges, called the vertex degree. For example, if every vertex in a graph is connected to 3 other vertices, it is a 3-regular graph. This uniformity is what makes regular graphs intriguing and beneficial for numerous practical applications, such as networking and circuit design.

Regular graphs ensure balance and symmetry in their structure. This is useful in scenarios where a uniformly distributed network or circuitry is necessary. Each vertex holding the same number of connections enables an even traffic or communication flow, minimizing bottlenecks.

• Each vertex in a regular graph has the same vertex degree.
• Examples include 3-regular (triangular mesh) or 4-regular (square lattice) graphs.
• Maintains symmetry which is often aesthetically pleasing and structurally sound for certain applications.

Cubic Graph

A cubic graph is a type of regular graph where every vertex connects to exactly three others, hence is also known as a 3-regular graph. This particular subclass is significant due to its applicability in designing balance within networks, and because of its unique geometric and structural properties.

When creating a cubic graph, it's essential to ensure that no vertex exceeds the degree of 3. This restriction often leads to interesting and unpredictable variations. Such graphs are pertinent in designing similar load distributed networks without overcomplicating the connections, making them ideal for certain types of network design like telecommunications.

• All vertices in a cubic graph have exactly three edges each.
• This structure is conducive to evenly distributing workload across a system.
• Examples include the cube, Petersen graph, and others that can serve various practical uses.

Graph Construction

Graph construction refers to the methods and processes used to shape suitable graphs for any application, under certain constraints or for specific properties. When constructing a graph, you can start with a basic framework and add or remove edges to meet particular requirements, like having a certain number of edges or ensuring every vertex fulfills certain conditions.

For various problems, such as those involving regular and connected graphs, graph construction allows you to create complex networks by adhering to degree-based protocols and maintaining connectivity. It's a trial-and-error process where understanding the essential properties guides the way.

• Graph construction involves adding/removing edges to obtain specific features.
• It's important to ensure the graph meets its intended properties, like regularity or connectivity.
• This process enables engineers and scientists to model practical structures efficiently.