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Whitney Sherman & Patti Bodkin
Saint Michael’s College
Definition
• Let e be an edge of a graph G that is neither a bridge nor a loop.
A bridge is an edge whose deletion separates the graph
A loop is an edge with both ends incident to the same vertex
H
bridgesNot a bridge A loop
G G
Graph Theory Terms
( ) | | ( )r G V k G The rank of a graph G is
( ) | | | | ( )n G E V k G The nullity of a graph G is
| |V is the number of vertices of G,
is the number of edges of G,
is the number of components of G. ( )k G
| |E
Example: | | 8E | | 8V ( ) 1k G
( ) 8 1 7r G ( ) 8 8 1 1n G
is the deletion of edgeG e e
t G x y t G e x y t G e x y; , ( ; , ) \ ; ,a f a f
Tutte Polynomial
/G e e is the contraction of edge
Deletion and Contraction
G-e
G/e
e
Delete e
Contract e
G
Deletion Contraction Method• If G consists of i bridges and j loops
• Then, where:
Example:
; , i jt G x y x y T ( ) = x , and T( ) = y.
+
= x2 + x + y= x2 + +
=
=
e
e
Universality of the Tutte Polynomial
( ) ( ) ( / )f G a f G e b f G e
( ) ( ) ( )f GH f G f H
fLet be a function of graphs such that:
e whenever is not a loop or an isthmus
GH G H where is either the disjoint union of and or where and share at most one vertexHG
| | | | ( ) | | ( ) 0 0( ) ( ; , )E V k G V k G x yf G a b t G
b a then,
| |E ( )k G where , , and are the number of edges, vertices, and components of respectively, and where
| |VG
f ( ) = x0 , and f( ) = y0.
Theorem:
Reliability Polynomial
Given an edge , which is not a loop or bridge, is defined ase ( ; )R G p
( ; ) (1 ) ( ; ) ( / ; )R G p p R G e p p R G e p
where is the probability that an edge in a network is workingp
and the probability that the edge is not working is 1 p
Recall from the universality theorem that
( ) ( ) ( / )f G a f G e b f G e e whenever is not a loop or an isthmus
For the reliability polynomial, and b p1a p
Reliability Polynomial
Recall also from the universality theorem that
( ) ( ) ( )f GH f G f H GH G H where is either the disjoint union of and or where and share at most one vertexHG
( ; ) ( ; ) ( ; )R GH p R G p R H p
0( ; )R p p x 0( ; ) 1R p y and
Reliability Polynomial
Since the reliability polynomial satisfies the two conditions of the universality theorem, we get:
| | | | ( ) | | ( ) 1( ; ) (1 ) ( ;1, )
1E V k G V k GR G p p p t G
p
The Reliability polynomial is an evaluation of the Tutte polynomial!!
Rank Generating Function
The rank generating function of the Tutte polynomial is defined as
( ) ( ) | | ( )( ; , ) ( 1) ( 1)r E r F F r F
F Et G x y x y
E V G where and are the sets of edges and vertices of the graph, , and is the rank of .( )r F F