1
Graphs
Discrete Mathematics
Graph
Definition:
• A simple graph G = (V, E) consists of V, a nonempty set
of vertices, and E, a set of unordered pairs of distinct
elements of V called edges.
•For each e∈E, e = {u, v} where u, v ∈V.
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oops.
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edge e is a loop if e = {u, u} for some u∈V.
v
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(u, v)
2
Simple Graph
• A simple graph is a finite undirected graph without loops
and multiple edges.
• A simple graph G=(V,E) consists of:
– a set V of vertices or nodes
– a set E of edges / arcs / links
3
Multigraphs
• A multigraph G = (V, E) consists of V, a nonempty finite
set of vertices and E, a finite multiset set of unordered
pairs of distinct elements of V called edges. Thus,
multiple edges between two vertices are allowed in a
multigraph.
4
Multiple edge
Multigraphs
• Edge-labels distinguish between edges sharing same
endpoints. Labeling can be thought of as function:
e1 Æ{1,2}
e2 Æ{1,2}
e3 Æ{1,3} 1 2
e1
e2
e4 Æ{2,3}
e5 Æ{2,3}
e6 Æ{1,2}
5
3 4
e3e4
e5
e6
Pseudographs
• If self-loops are allowed we get a pseudograph. Edges
may be associated with a single vertex, when the edge is
a loop
e1 Æ{1,2}
e2 Æ{1,2}
3
Æ
{1 3}
e1
e6
e
3
Æ
{1
,
3}
,
e4 Æ{2,3}
e5 Æ{2}
e6 Æ{2}
e7 Æ{4}
6
1 2
3 4
e3
e2
e4
e5e7
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