Graphs CSCI 2720 Spring 2005. Graph Why study graphs? important for many real-world applications...
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Transcript of Graphs CSCI 2720 Spring 2005. Graph Why study graphs? important for many real-world applications...
Graph
Why study graphs? important for many real-world
applications compilers Communication networks Reaction networks & more
The Graph ADT a set of nodes (vertices or points) connection relations (edges or arcs)
between those nodes
Definitions follow ….
Definition : graph A graph G=(V,E) is
a finite nonempty set V of objects called vertices (the singular is vertex)
together with a (possibly empty) set E of unordered pairs of distinct vertices of G called edges.
Some authors call a graph by the longer term ``undirected graph'' and simply use the following definition of a directed graph as a graph. However when using Definition 1 of a graph, it is standard practice to abbreviate the phrase ``directed graph'' (as done below in Definition 2) with the word digraph.
Definition: digraph A digraph G=(V,E) is
a finite nonempty set V of vertices together with a (possibly empty) set E of
ordered pairs of vertices of G called arcs.
An arc that begins and ends at a same vertex u is called a loop. We usually (but not always) disallow loops in our digraphs.
By being defined as a set, E does not contain duplicate (or multiple) edges/arcs between the same two vertices.
For a given graph (or digraph) G we also denote the set of vertices by V(G) and the set of edges (or arcs) by E(G) to lessen any ambiguity.
Definition: order, size The order of a graph (digraph)
G=(V,E) is |V|, sometimes denoted by |G| , and the size of this graph is |E| .
Sometimes we view a graph as a digraph where every unordered edge (u,v) is replaced by two directed arcs (u,v) and (v,u) . In this case, the size of a graph is half the size of the corresponding digraph.
Example G1 is a graph of order 5 G2 is a digraph of order
5 The size of G1 is 6
where E(G1) = {(0, 1), (0, 2), (1, 2), (2,
3), (2, 4), (3, 4)} The size of the digraph
G2 is 7 where E(G2) = {(0, 2), (1, 0), (1, 2), (1,
3), (3, 1), (3, 4), (4, 2)}.
Definition: walk, length, path, cycle A walk in a graph (digraph) G is
a sequence of vertices v0, v1, … vn such that, for all 0 <= i< n , (vi, vi+1) is an edge (arc) in G .
The length of the walk v0, v1, … vn is the number n (i.e., number of edges/arcs).
A path is a walk in which no vertex is repeated.
A cycle is a walk (of length at least three for graphs) in which v0
=vn and no other vertex is repeated; sometimes, if it is understood, we omit vn from the sequence.
Definition: connected, strongly connected
A graph G is connected if there is a path between all pairs of
vertices u and v of V(G) . A digraph G is strongly connected
if there is a path from vertex u to
vertex v for all pairs u and v in V(G).
Definition: degree In a graph, the degree of a vertex v ,
denoted by deg(v), is the number of edges incident to v . in-degree == out-degree
For digraphs, the out-degree of a vertex v is the number of arcs {(v,z) € E| z € V}
incident from v (leaving v ) and the in-degree of vertex v is the number of arcs {(z,v) € E| z € V} incident to v (entering v ).
Degree, degree sequence G1:
deg(0) = 2 deg(1) = 2 deg(2) = 4 deg(3) = 2 Deg(4) = 2
Degree sequence = (2,2,4,2,2)
Degree, degree sequence G2:
In-degree sequence = (1,1,3,1,1)
Out-degree sequence = (1,3,0,2,1)
Degree of vertex of a digraph sometimes written as sum of in-degree and out-degree:
(2,4,3,3,2)
Definition: diameter
The diameter of a connected graph or strongly connected digraph G=(V,E) is the least integer D such that for all
vertices u and v in G we have d(u,v) <=D, where d(u,v) denotes the distance from u to v in G, that is, the length of a shortest path between u and v.
Computer representations adjacency matrices
For a graph G of order n , an adjacency matrix representation is a boolean matrix (often encoded with 0's and 1's) of dimension n such that entry (i,j) is true if and only if edge/arc (I,j) is in E(G).
adjacency lists For a graph G of order n , an adjacency lists
representation is n lists such that the i-th list contains a sequence (often sorted) of out-neighbours of vertex i of G .
Matrix vs. list representation Matrix
n vertices and m edges requires O( n2 ) storage check if edge/arc (i,j) is in graph – O(1)
List n vertices and m edges, requires O(m) storage Preferable for sparse graphs tcheck if edge/arc (i,j) is in graph - O(n) time
Note: other specialized representations exist