Basic Concepts Graphs For more notes and topics visit:

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Presentation transcript:

Basic Concepts Graphs For more notes and topics visit: www.eITnotes.com eITnotes.com

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 eITnotes.com

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. eITnotes.com

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 eITnotes.com

Example Vertices: A,B,C,D Edges: AB, AC, BC, CD A C B D A B C D Graph1 eITnotes.com Two ways to draw the same graph

Directed and undirected graphs B C D eITnotes.com These are two different graphs

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 eITnotes.com

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 eITnotes.com

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 eITnotes.com A graph without cycles is called acyclic graph.

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 eITnotes.com

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 eITnotes.com

Complete graphs Graphs with all edges present – each vertex is connected to all other vertices A B C D E A complete graph eITnotes.com

Weighted graphs Weighted graphs – weights are assigned to each edge (e.g. road map) B 2 1 C A 2 4 3 D eITnotes.com

Graph Representation Adjacency matrix Adjacency lists eITnotes.com

Adjacency matrix – undirected graphs Vertices: A,B,C,D Edges: AC, AB, AD, BD A B C D A 0 1 1 1 B 1 0 0 1 C 1 0 0 1 D 1 1 0 0 The matrix is symmetrical A B C D eITnotes.com

Adjacency matrix – directed graphs Vertices: A,B,C,D Edges: AC, AB, BD, DA A B C D A 0 1 1 0 B 0 0 0 1 C 0 0 0 0 D 1 0 0 0 A B C D eITnotes.com

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 eITnotes.com

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 eITnotes.com