degree

                                                               DEGREE OF A GRAPH

                       Number of edges incident with a graph is called degree of a graph.

                                In this graph every vertex is labelled by its degree.An isolated vertex is having degree 0.

                                If the degree of every vertex is same that graph is called regular graph.eg.Kregular graph,Pregular graph.
This is a regular graph because each and every vertex is having degree 3.
                                                            Cycles and complete graphs are always regular graphs.

                                                           HANDSHAKING THEOREM


                                                             Sum of the degrees of all the vertices of a graph is twice the number of edges.This theorem is known as handshaking theorem.
                                                             In the example the sum of edges is 5 and the sum of degrees of the edges is 10.so sum of the degrees of all the vertices of this graph is twice the number of edges.

                                                                DEGREE SEQUENCE

                                                              It is a sequence of non-negative integers which would be the of vertices of some graph.



                                                                     GRAPH INVARIANTS

1. Number of vertices
2. Number of edges.
3. Degree sequence
4. Number of cycles
5.  Number of paths

                                                               COMPLEMENT OF A GRAPH

The complement of a graph with vertex set V denoted by G' is another graph with same vertex set V with the condition that 2 vertices are adjacent in G if and only if they are not adjacent  in G'

                                                The combination of a graph and its complement is a complete graph.

representation of graphs

                                             ADJACENCY MATRIX AND GRAPHS 

Any graphs can be represented as adjacency matrices without even drawing the vertices and edges.
                                                                                 



Here the graph itself is represented using the adjacency matrix.Vertices connected to each other by edges are denoted as 1 and otherwise 0.This matrix is enough to identify the graph.

                                                         INCIDENCE RELATION


                       



                               Entry of an incidence matrix is set to 1 if an edge incidence on a vertex otherwise it is set to zero.


                                                            ADJACENCY LIST



                          In this representation,every vertexes connected to a particular edge is plotted.

classification of graphs

                               DEEPER CLASSIFICATION OF GRAPHS

Here we are classifying the graphs by considering the graphs as a set of edges and vertices.E denotes the edge set and V denotes the vertex set.

• null graph

                                                    In a null graph none of the objects are related to each other,then the graph produced out of it is called a null graph.

• Pseudo graph

                                                       Here we can see that two vertexes are connected to itself.So in graphs in which if any vertex is connected to itself is called pseudo graph.Such an edge is called self loop. A graph

with self loop is called pseudo graph.

• Multi-graph

  If in any graph between two vertices there exists more than 1 relation,they are connected by more than 1 edge;such edges are called parallel edges.if in any graph there exists atleast one pair of parallel edges,such a graph is called multi-graph.

• Simple graph

If any graph has neither a self loop nor even a pair of parallel edges is called a simple graph.

                                                 Simple graphs can be given different names depending on the structure of the graphs.

1. complete graphs

                                   If every vertex of a simple graph is connected to all other vertices ,then that graph is known as complete graph. It  is usually denoted as k1(complete graph with i vertex),k2(complete graph with 2 vertices)..................

     2.   cycle graph 

                                If every vertex is connected to exactly two other vertices then that graph is called as a cycle graph.

It is denoted as Cn. Its first value is 3.

   3.  Path graph

                               It is a continuous sequence of vertices and edges starting from a vertex exactly at another vertex.Denoted by Pn.

   4.  star graph

                                       

   It is a graph where exactly 1 vertex is connected to each other vertex,denoted by the symbol Sn.

  5.Wheel graph

                                                 A vertex is connected to every vertex of a cycle the resultant graph is called a wheel graph denoted by Wn. Its starts from 4.