Graph Theory Concepts

Definition of a semi-Eulerian graph

• A connected graph is semi-Eulerian if there is an open trail that includes every edge once only.

• A connected graph is semi-Eulerian if it contains a cycle that visits every vertex exactly once.

• A connected graph is semi-Eulerian if it has at least one vertex of odd degree.

• A connected graph is semi-Eulerian if it can be divided into two disjoint subgraphs.

Path

• A sequence of edges that connect a sequence of vertices.

• A collection of vertices without any edges.

• A single edge connecting two vertices.

• A loop that returns to the starting vertex.

Definition of a semi-Hamiltonian graph

• A semi-Hamiltonian graph contains a path that includes every vertex once only but is not a cycle.

• A semi-Hamiltonian graph is a graph that contains a cycle visiting every vertex exactly once.

• A semi-Hamiltonian graph is a complete graph with all vertices connected in a cycle.

• A semi-Hamiltonian graph is a graph that has no cycles at all.

Definition of a loop in a graph

• A loop is an edge in a graph that joins a vertex to itself.

• A loop is a cycle that connects two different vertices.

• A loop is a path that traverses multiple edges in a graph.

• A loop is a vertex that has no edges connected to it.

Number of faces in the given planar graph

• 2

• 3

• 4

• 5

Difference between Hamiltonian and semi-Hamiltonian graphs

• A Hamiltonian graph forms a cycle, while a semi-Hamiltonian graph does not.

• Both Hamiltonian and semi-Hamiltonian graphs form cycles.

• A Hamiltonian graph has no cycles, while a semi-Hamiltonian graph does.

• Semi-Hamiltonian graphs are always complete graphs.

Definition of a planar graph

• A graph that can be drawn on a plane without any edges crossing.

• A graph that contains at least one cycle.

• A graph that can be represented in three-dimensional space.

• A graph where all vertices are connected to each other.

Tree

• A connected graph with no cycles.

• A disconnected graph with cycles.

• A graph with multiple edges between the same vertices.

• A graph that contains at least one cycle.

Network

• A system used for data storage and retrieval.

• A system often used in everyday life, such as television or rail networks, and can represent trails, social networks, transport networks, food webs, and sporting competitions.

• A type of computer program for managing files.

• A collection of unrelated items.

Characteristics of a bipartite graph

• In a bipartite graph, vertices are divided into two groups with edges only between groups.

• A bipartite graph contains cycles of odd length.

• All vertices in a bipartite graph are connected to every other vertex.

• A bipartite graph has at least one vertex with degree zero.