Exploring Graph Theory Fundamentals

What is a graph in graph theory?

• A graph is a collection of vertices and edges connecting them.

• A graph is a linear arrangement of numbers.

• A graph consists only of edges without vertices.

• A graph is a type of mathematical function.

Define the terms 'vertex' and 'edge'.

• Vertex: a point in a graph; Edge: a connection between two vertices.

• Vertex: a shape in geometry; Edge: a boundary of a shape.

• Vertex: a connection between two edges; Edge: a point in a graph.

• Vertex: a line in a graph; Edge: a point in a network.

What is the difference between a directed and an undirected graph?

• The main difference is that directed graphs have edges with direction, while undirected graphs have edges without direction.

• Directed graphs can only have one edge between two vertices, while undirected graphs can have multiple edges.

• Undirected graphs are always weighted, while directed graphs are not.

• Directed graphs can have cycles, while undirected graphs cannot.

Explain what a weighted graph is.

• A weighted graph is a graph where all edges are equal in length.

• A weighted graph is a type of graph that does not allow loops or multiple edges.

• A weighted graph is a graph that only contains vertices without edges.

• A weighted graph is a graph where each edge has a numerical value (weight) associated with it.

What is a complete graph?

• A complete graph has no edges between vertices.

• A complete graph is a graph where every pair of vertices is connected by an edge.

• A complete graph is a graph where vertices are connected in a circular manner.

• A complete graph is a graph with only one vertex.

Define a bipartite graph.

• A bipartite graph is a graph with only one set of vertices.

• A bipartite graph is a graph with all vertices connected to each other.

• A bipartite graph is a graph whose vertices can be divided into two disjoint sets with edges only between the sets.

• A bipartite graph is a graph that contains no edges at all.

What is the degree of a vertex?

• The degree of a vertex is the total number of vertices in the graph.

• The degree of a vertex is the length of the longest edge connected to it.

• The degree of a vertex is the number of faces that share the vertex.

• The degree of a vertex is the number of edges incident to it.

Explain what a path is in a graph.

• A path is a sequence of vertices connected by edges in a graph.

• A path is a collection of edges without vertices.

• A path is a single vertex in a graph.

• A path is a random selection of nodes in a graph.

What is a cycle in graph theory?

• A cycle is a path that can only be traversed in one direction.

• A cycle is a path that connects two different vertices without returning to the start.

• A cycle is a closed path in a graph where the starting and ending vertex is the same, with no other vertices repeated.

• A cycle is a sequence of vertices where all vertices are distinct.

What does it mean for a graph to be connected?

• A graph is connected if there is a path between every pair of vertices.

• A graph is connected if it contains no cycles.

• A graph is connected if it has at least one vertex.

• A graph is connected if it has an even number of edges.