The document defines several graph terminology terms including adjacent vertices, which are vertices connected by an edge. It also defines predecessor and successor vertices for directed graphs, degree of a vertex as the number of edges incident to it, and in-degree and out-degree for directed graphs. As an example, it calculates the in-degrees, out-degrees, and degrees of the vertices in a sample directed graph.

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Graph Terminology :
Adjacent vertices :
An adjacent vertex of a vertex v in a graph G is a vertex
that is connected to v by an edge.
( or )
In simple if two vertices in an undirected graph are
connected by an edge, then they are called as Adjacent vertices
or Neighbors.
Fig 1
In fig1:
Adjacent vertices of a : b,c,e
Adjacent vertices of b : a
Adjacent vertices of c : a,d,e
Adjacent vertices of d : c,e
Adjacent vertices of e : a,c,d

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 But For an directed graph it is different .For example
Fig: 2
 Here there is an edge between 3, 2 which is represented as
(3, 2) means 2 is adjacent to 3 but 3 is not adjacent to 2.
Predecessor and Successor :
The vertex of which the arrow comes out is called
Predecessor and the vertex that is pointing by the arrow is
called Successor.
In Fig 2. Vertex 3 is the Predecessor and Vertex 1 is the
Successor.
Degree of a Vertex :
In graph theory, the Degree of a vertex of a graph is the
Number of edges that are incident on that vertex.
 If there is a loop, the edges of that loop is counted as twice.
 The degree of a vertex v is denoted deg(v) or deg v.

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 The maximum degree of a graph G, denoted by Δ(G)
 The minimum degree of a graph, denoted by δ(G)
Fig 3
In the above fig.3, the degree of vertex 5 is 5
Terminology on vertex :
 A vertex with degree 0 is called an Isolated vertex.
In the above fig.3, The degree of vertex 0 is 0.
Fig .4

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 A vertex with degree 1 is called a Leaf vertex or End
vertex, and the edge incident with that vertex is called a
Pendant edge. In the graph (fig 4) on the right, {3,5} is a
pendant edge.
 A vertex with degree n − 1 in a graph on n vertices is called
a Dominating vertex.
In-degree and Out-degree :
In a directed graph it is important to distinguish between
In-degree and Out-degree. For any directed edge, it has two
distinct ends: a head (the end with an arrowhead) and a tail.
Each end is counted separately. The sum of head endpoints
count toward the In-degree of a vertex and the sum of tail
endpoints count toward the Out-degree of a vertex.
 The In-degree of a vertex V written by deg −(v).
 The Out-degree of a vertex V written by deg +(v)
For example :

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Fig .5
 vertex 2 has In-degree 2.
 vertex 2 has Out-degree 1.
Question :
Find the In -Degree, Out-degree, and degree of each vertex of a
graph given below.
Fig.6
In-Degree of a vertex 'v1' = deg(v1) = 1 and Out-Degree of a
vertex 'v1' = deg(v1) = 2
In-Degree of a vertex 'v2' = deg(v2) = 1 and Out-Degree of a

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vertex 'v2' = deg(v2) = 3
In-Degree of a vertex 'v3' = deg(v3) = 1 and Out-Degree of a
vertex 'v3' = deg(v3) = 2
In-Degree of a vertex 'v4' = deg(v4) = 5 and Out-Degree of a
vertex 'v4' = deg(v4) = 0
In-Degree of a vertex 'v5' = deg(v5) = 1 and Out-Degree of a
vertex 'v5' = deg(v5) = 2
In-Degree of a vertex 'v6' = deg(v6) = 0 and Out-Degree of a
vertex 'v6' = deg(v6) = 0
 By the definition, the degree of a vertex is Deg(v) = deg−(v)
+ deg+(v). Therefore
Degree of a vertex 'v1' = deg(v1) = 1 + 2 = 3
Degree of a vertex 'v2' = deg(v2) = 1 + 3 = 4
Degree of a vertex 'v3' = deg(v3) = 1 + 2 = 3
Degree of a vertex 'v4' = deg(v4) = 5 + 0 = 5
Degree of a vertex 'v5' = deg(v5) = 1 + 2 = 3
Degree of a vertex 'v6' = deg(v6) = 0 + 0 = 0