Graphs
• A graph is a set of objects called vertices connected by lines
called edges. Each edge connects two vertices and represents a
relationship or link between them.
The individual entities or points in the graph.
The connections or relationships between the
vertices
A graph with Finite Vertices and Edges.
A graph with infinte Vertices or Edges.
A graph in which two edges are connects to
two different Vertices (no Multiple edges or loop).
A graph in which two edges are not
connected to two different Vertices (there can be a loop
too).
An Edge that connects a vertex to itself.
• Simple Graph:
• Multiple Graph:
A Graph in which multiple edge and loops are involved.
A Graph with multiple edges or loops.
A Multiple graph with loops.
A graph in which each edge
is associated with ordered pair
of vertices.
A graph in which each edge
is associated with unordered
pair of vertices.
A Graph in which some edges are directed and some are
undirected.
• Number of edges connected to a Graph.
The degree of a vertex 𝑣 is often denoted as deg( 𝑣).
All vertices have deg(2):
In directed graphs, the in-degree is the number of
edges coming into the vertex.
In directed graphs, the out-degree is the number
of edges leaving the vertex
A vertex with a degree of 0 (no edges
connected).
A vertex with a degree of 1 (one edge
connected).
A graph where every vertex has the same
degree.
In any undirected graph, the sum of the degrees of all
vertices is equal to twice the number of edges.
• V is the set of vertices deg(𝑣)
• deg(v) is the degree of vertex 𝑣
• E is the number of edges in the graph
A null graph is a graph with
no edges. It is also known
as an empty graph.
There can be multiple
vertices but degree of
vertices will be zero.
A cyclic graph is a graph
that has at least one cycle.
A cycle in a graph is a path
that begins and ends at the
same vertex and visits at
least one other vertex
exactly once.
Each vertex is connected to
all of the other vertices.
The number of edges in a
complete graph with 𝑛
vertices is given by the
formula:
A wheel graph is a graph
that consists of a cycle with
n−1 vertices and a central
vertex connected to every
vertex on the cycle.
The center Vertex is not
counted in the number of
total vertices.
Cubic Graph (also known
as Hypercube) is a graph
that has vertices
representing the 2n bit
string of length n.
A Bipartite graph is a type
of graph where the vertices
can be divided into two
disjoint sets such that no
two vertices within the
same set are adjacent.
The union of two graphs G and
H is the union of their vertices
and their edges:
The Intersection of two graphs G
and H is the intersection of their
vertices and their edges.
The complement of a graph is a graph that has same
number of vertices but the edges that are connected in G
are not connected
in and vice versa.
A graph G is equal to G(v , e) and its subgraph H(w , f) is
obtained by eliminating its vertex/vertices and the edges
connecting it.
W
V
F
E
Number of subgraph with atleast one vertex:
(2e - 1)
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