This document defines basic concepts in graph theory. A graph consists of a set of vertices and edges connecting pairs of vertices. An adjacency matrix represents which vertices are connected by edges, while an incidence matrix represents which edges connect to each vertex. A simple graph cannot have loops or multiple edges between vertices. A complete graph connects each pair of vertices. Two graphs are isomorphic if there is a one-to-one correspondence between their vertices that preserves edge connections. Multigraphs allow multiple edges between vertices, while pseudographs also allow loops. The degree of a vertex is the number of edges connected to it.

1. GRAPH THEORY
MRS.D.ARIVUSELVI
,M.SC. ,B.Ed. ,M.Phil

2. BASIC DEFINITIONS
GRAPH:
A graph G = (V (G),E(G)) consists of a non-empty finite set V (G) of
elements called vertices, and a finite set E(G) of unordered pairs of
elements of V (G) called edges. We call V (G) the vertex set and E(G) the
edge set of G. An edge {v, w} is said to join the vertices v and w, and is
usually abbreviated to vw.
For example,
This diagram represents the graph G whose vertex set V (G) is {u, v,w, z},
and whose edge set E(G) consists of the edges uv, uw, vw and wz. The
numbers of elements in V (G) and E(G) are denoted by |V (G)| and |E(G)|
respectively

3. ADJACENCY AND INCIDENCE MATRIX:
If G is a graph with vertices unlabelled {1, 2,…, n}, its adjacency matrix
A(G) is the n × n matrix whose ijth entry is the number of edges joining
vertex i and vertex j. If, in addition, the edges are labelled {1, 2,…,m}, its
incidence matrix M(G) is the n ×m matrix whose ijth entry is 1 if vertex i is
connected to edge j; and 0 otherwise. Figure 4 shows a labelled graph G
with its adjacency matrix A and incidence matrix respectively.
SIMPLE GRAPH:
A Simple graph is a graph that has no loops and no multiple edges

4. COMPLETE GRAPH
A simple graph in which each pair of distinct vertices are adjacent is a
complete graph. We denote the complete graph on n vertices by Kn. K3 and
K4
K3 K4
ISOMORPHIC:
Let two graphs G1 and G2 are isomorphic (written as G1 ≃ G2) if there is a
one-one correspondence between the vertices of G1 and those of G2 such
that the number of edges joining any two vertices of G1 is equal to the
number of edges joining the corresponding vertices of G2. Let θ : V (G1) →
V (G2) be the isomorphism, then x1 x2 E (G1) if and only if θ(x1)θ(x2)
E(G2).

5. Multigraph
If more than one line joining two vertices are allowed
,the resulting object is called a multigraph
Pseudo Graph
If further loops are allowed ,the resulting object is
called a Pseudo graph.
Null Graph
A Graph whose edge set is empty is called a Null
Graph

6. Degree
The degree of a point vi in a graph G is the
number of lines incident with vi .The degree
of vi is denoted by degvi.
Isolated Point
A point v of degree 0 is called an
isolated Point.
End point
A point v of degree 1 is called an end point.

7. Acyclic Graph
A Graph that contains no cycles is called
an Acyclic graph.
Tree
A connected acyclic graph is called a tree.

9. Thank you