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