Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Definition: Two vertices u and v in an undirected graph G are adjacent (or
neighbors) in G if u and v are endpoints of an edge e in G. Edge e is incident to
{u,v} and e connects u and v.
Definition: The degree of a vertex v, denoted deg(v), in an undirected graph is
the number of edges incident with it except that loops contribute twice to the
degree of that vertex. If deg(v) = 0, then it is isolated. If deg(v) = 1, then it is a
pendant.
Handshaking Theorem: If G = (V,E) is an undirected graph with e edges, then
#
&
e= % deg(v) ( /2.
$
" v!V
'
Proof: Each edge contributes 2 to the sum since it is incident to 2 vertices.
Example: Let G = (V,E). Suppose |V| = 100,000 and deg(v) = 4 for all v,V.
Then there are (4$100,000)/2 = 200,000 edges.
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Theorem: An undirected graph has an even number of vertices and an odd
degree.
Definition: Let (u,v),E in a directed graph G(V,E). Then u and v are the initial
and terminal vertices of (u,v), respectively. The initial and terminal vertices of a
loop (u,u) are both u.
Definition: The in-degree of a vertex, denoted deg 4 (v), is the number of edges
with v as their terminal vertex. The out-degree of a vertex, denoted deg + (v), is
the number of edges with v as their initial vertex.
Theorem: For a directed graph G(V,E), # deg ! (v) = deg
v"V # + (v) = E .
v"V
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