We study here some type of graph like multigraph ,multigraph,path,circuit,etc...
now we define here a graph. start as...
DIRECTED GRAPH:- A directed graph is a graph defined abstractly as an ordred pair (V,E) where V={v1,v2,.....} vertex and E={e1,e2,e3.........} i.e. egdes is a binary relation on V. The pair (vi,vj) is said to be incedent vertices.where vi is a initial vertax and vj is a terminal vertix.
SELF LOOP:- The defination of a graph an edge to be of the form (vi,vj) suvh an edge having the same vertex as both its end vertices is called a self loop
where K is a self loop
PARALLEL EDGES IN A GRAPH:- Let G=(V,E) is a graph then all edges having the same pair of end vertices are called parallel edges ..
where e1 and e2 are parallel edges.
UNDIRECTED GRAPH:- An undirected graph is defined abstractly as an ordered pair (V,E) where V is a non-empty set and E is a multisets of two elements from V.for example
this is an undirected graph because it have not any direction.
SIMPLE GRAPH:- A graph G=(V,E) that has neither self -loop nor parallel edges is called a simple graph. for example
it have no self loop and no parallel edges.
FINITE AND INFINITE GRAPH:- A graph with a finite number of vertices as well as a finite number of edges is called a finite graph otherwise it is infinite graph.
this is a finite graph because it have finite number of vertices and edges.
this is an infinite graph.
ORDER OF A GRAPH:- If G=(V,E) is a fiinite group then the number of vertices is is called the order of the graph G and its denoted by IVI (mod V).
INCEDENCE :- Let E be and edge joining two vertices vi to vj of a graph G=(V,E) then the edge e is said to be incident on each of its end vertices vi to vj.
ADJACENCY:- Two vertices in a graph are said to ve adjacent if there are exist an edge joining the vertices.
DEGREE OF A VERTEX:- The degree of a vertex v in a graph written as d(v) is equal to the number of edges which are incident on v with self -loop counted twice.
In fig A d(a)=2 , d(b)=2 , d(C)= 2
ISOLATED VERTEX:- a vertex in a graph G having no edge incedent on it is called an isolated vertex.
in this graph f is a isolated vertex.
PENDANT VERTEX:- A vertex v in a graph G is said to ve pendent vertex if its degree is one. i.e. d(v) = 1 . in fig B degree of e is 1.
NULL GRAPH:- A graph is said to be null if its each vertex is zero degree or other word its vertices are non empty but its edes are empty.
EVEN OR ODD VERTICES:- A vertex is said to be even or odd if its degree is an even of odd number.
THEORAM:- The sum of the degrees of all vertices in a graph G is equal to twice the number of edges in G.
solve:- Let G=(V,E) be a graph ,then the number of edges in Gis IEI,
since the edge in G is incident on two vertices so it contributes 2 to the sum of the degrees of the graph ,
so sum of degrees of all the vertices in G = ∑d(v)=2IEI
this theoram is called the HANDSHAKING LEMMA.
FOR EXAMPLE:-
now we define here a graph. start as...
DIRECTED GRAPH:- A directed graph is a graph defined abstractly as an ordred pair (V,E) where V={v1,v2,.....} vertex and E={e1,e2,e3.........} i.e. egdes is a binary relation on V. The pair (vi,vj) is said to be incedent vertices.where vi is a initial vertax and vj is a terminal vertix.
SELF LOOP:- The defination of a graph an edge to be of the form (vi,vj) suvh an edge having the same vertex as both its end vertices is called a self loop
where K is a self loop
PARALLEL EDGES IN A GRAPH:- Let G=(V,E) is a graph then all edges having the same pair of end vertices are called parallel edges ..
UNDIRECTED GRAPH:- An undirected graph is defined abstractly as an ordered pair (V,E) where V is a non-empty set and E is a multisets of two elements from V.for example
SIMPLE GRAPH:- A graph G=(V,E) that has neither self -loop nor parallel edges is called a simple graph. for example
 |
| FIG A |
FINITE AND INFINITE GRAPH:- A graph with a finite number of vertices as well as a finite number of edges is called a finite graph otherwise it is infinite graph.
this is a finite graph because it have finite number of vertices and edges.
ORDER OF A GRAPH:- If G=(V,E) is a fiinite group then the number of vertices is is called the order of the graph G and its denoted by IVI (mod V).
INCEDENCE :- Let E be and edge joining two vertices vi to vj of a graph G=(V,E) then the edge e is said to be incident on each of its end vertices vi to vj.
ADJACENCY:- Two vertices in a graph are said to ve adjacent if there are exist an edge joining the vertices.
DEGREE OF A VERTEX:- The degree of a vertex v in a graph written as d(v) is equal to the number of edges which are incident on v with self -loop counted twice.
In fig A d(a)=2 , d(b)=2 , d(C)= 2
ISOLATED VERTEX:- a vertex in a graph G having no edge incedent on it is called an isolated vertex.
 |
| fig B |
PENDANT VERTEX:- A vertex v in a graph G is said to ve pendent vertex if its degree is one. i.e. d(v) = 1 . in fig B degree of e is 1.
NULL GRAPH:- A graph is said to be null if its each vertex is zero degree or other word its vertices are non empty but its edes are empty.
THEORAM:- The sum of the degrees of all vertices in a graph G is equal to twice the number of edges in G.
solve:- Let G=(V,E) be a graph ,then the number of edges in Gis IEI,
since the edge in G is incident on two vertices so it contributes 2 to the sum of the degrees of the graph ,
so sum of degrees of all the vertices in G = ∑d(v)=2IEI
this theoram is called the HANDSHAKING LEMMA.
FOR EXAMPLE:-
an element c∈B st (a,c)∈R , (c,b)∈S}