Graph Theory Flashcards
This deck is an introduction to graph theory.
What is a graph?
Object consisting of 2 sets called Vertex set and Edge set
What is Vertex set?
Finite nonempty set
Example:
{P. Q, R, S, T, U}
What is Edge set?
Pair of vertex elements. It can be empty.
{{P, Q}, {P, R}, {Q, R}, {S, U}}
Vertices and Edges
The elements of the vertex set are called vertices and the elements of the edge set are called edges.
v - number of vertices
e - number of edges
What are adjacent vertices
Assume {X, Y} is edge of a graph {X, Y} connects the vertices X and Y together. X and Y are adjacent to one another.
What are incident edges
The edge {X, Y} is incident to each X and Y and each of X and Y is incident to {X, Y}
What are adjacent Edges
Two edges incident to the same vertex.
What is Isolated Vertex?
A vertex incident to no edge.
What are graph diagrams?
Representation of a graph. The vertices are represented by heavy dots. Vertex adjacency is represented by connecting the corresponding dots.
What is graph equality?
Two graphs are equal if the have equal vertex sets and equal edge sets.
What is graph diagram equality?
Two graph diagrams are equal if they represent equal vertex set and equal edge set.
Example:
{P, Q, R, S, T, U}
{{P, Q}, {P, R}, {Q, R}, {S, U}}
{T, P, R, Q, U, S}
{{Q, R}, {R, P}, {U, S}, {P, Q}}
What are undirected graphs?
The edges of a graph are undirected.
{A, B} = {B, A}
What are Cyclic graphs?
Graphs having v>=3 and vertex set {1, 2, 3, … v} and edge set {{1, 2}, {2, 3}, {3, 4}, ….{v, 1}}. They are denoted by Cv
\_\_\_\_\_\_\_\_\_ | | | | | | ------------------
What are Complete graphs?
Graph having all possible edges. Denoted by Kv
What are Null graphs?
Graphs having vertex set but not an edge set. Denoted by N
What is utility graph?
Graph having vertex set {A, B, C, X, Y, Z} and edge set
{{A, X}, {A, Y}, {A, Z}, {B, X}, {B, Y}, {B, Z}, {C, X}, {C, Y}, {C, Z}}.
Unlike other types of graph Utility Graph is only one. Denoted by UG.
What is the formula to find how many edges a utility graph has?
e = 1/2v(v-1)
What is complement of graphs?
Complement of graph G denoted by G’ is a graph with same vertices bu entirely different edges. If two vertices are connected in G they are not connected in G’ and vice versa.
Null graphs and Complete graphs are complementary. For every positive integer v, Nv’ is equal to Kv and Kv’ is equal to Nv
What are subgraphs?
A graph H is a subgraph of G when the vertex set of H is a subset of G and the edge set of H is a subset of Gs edge set.
Every graph is a subset of themselves.