1. Basic Concepts Graphs For more notes and topics visit:
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2. Graphs – Basic Concepts
Basic definitions: vertices and edges
More definitions: paths, simple paths, cycles, loops
Connected and disconnected graphs
Complete graphs
Weighted graphs
Graph representation
Adjacency matrix
Adjacency lists
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3. Basic Graph Definitions
A graph is a non-linear data structure that consists of a collection of nodes(or vertices) and a collection of edges(or arcs), with each edge joining one node to another.
Formally, a graph G = (V,E) consists of two sets:
V called set of all vertices or nodes.
E called set of all edges or arcs.
Vertices are represented by circles or points and edges as line segments or arcs connecting the vertices.
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4. Vertices and Edges
Definition: A graph is a collection (nonempty set) of vertices and edges
Vertices: can have names
Edges: connect two vertices,
can be labeled,
can be directed
Adjacent vertices: there is an edge between them
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5. Example Vertices: A,B,C,D Edges: AB, AC, BC, CD A C B D A B C D Graph1
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Two ways to draw the same graph

6. Directed and undirected graphsB C D
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These are two different graphs

7. More definitions : Path
A list of vertices in which successive vertices are connected by edges
A B C
B A C D
A B C A B C A B C D
B A B A C A B C D
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8. More definitions : Simple Path
No vertex is repeated.
A B C D
D C A
D C B
A B
A B C A B C D
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9. More definitions : Cycle
A loop is a special case of cycle in which an edge begin and ends with the same vertex. A B C D
A B C A
B A C B
C B A C
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A graph without cycles is called acyclic graph.

10. Connected and Disconnected graphs
Connected graph: There is a path between each two vertices
Disconnected graph : There are at least two vertices not connected by a path.
Examples of disconnected graphs: A B C D
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11. Degree of vertex
In an undirected graph, the degree of a vertex is the number of edges originating from it.
In directed graph degree of vertex comes in two variations:
a) Indegree
b) Outdegree
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12. Complete graphs
Graphs with all edges present – each vertex is connected to all other vertices A B C D E
A complete graph
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13. Weighted graphs
Weighted graphs – weights are assigned to each edge (e.g. road map) B 2 1 C A 2 4 3 D
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14. Graph Representation
Adjacency matrix
Adjacency lists
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15. Adjacency matrix – undirected graphs
Vertices: A,B,C,D
Edges: AC, AB, AD, BD
A B C D
  A B C D
The matrix is symmetrical A B C D
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16. Adjacency matrix – directed graphs
Vertices: A,B,C,D
Edges: AC, AB, BD, DA
A B C D  A B C D A B C D
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17. Adjacency lists – undirected graphs
Vertices: A,B,C,D
Edges: AC, AB, AD, BD
Heads lists
A B C D
B A D
C A
D A B A B C D
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18. Adjacency lists – directed graphs
Vertices: A,B,C,D
Edges: AC, AB, BD, DA
Heads lists
A B C
B D
C =
D A A B C D
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