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 bridges Not a bridge A loop G G

Graph Theory Terms The rank of a graph G is The nullity of a graph G is is the number of vertices of G, is the number of edges of G, is the number of components of G. Example:

is the deletion of edge Tutte Polynomial 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: T ( ) = x, and T( ) = y. + = x 2 + x + y = x = = e e

Universality of the Tutte Polynomial Let be a function of graphs such that: whenever is not a loop or an isthmus where is either the disjoint union of and or where and share at most one vertex then, where,, and are the number of edges, vertices, and components of respectively, and where f ( ) = x 0, and f( ) = y 0. Theorem:

Reliability Polynomial Given an edge, which is not a loop or bridge, is defined as where is the probability that an edge in a network is working and the probability that the edge is not working is Recall from the universality theorem that whenever is not a loop or an isthmus For the reliability polynomial, and

Reliability Polynomial Recall also from the universality theorem that where is either the disjoint union of and or where and share at most one vertex and

Reliability Polynomial Since the reliability polynomial satisfies the two conditions of the universality theorem, we get: The Reliability polynomial is an evaluation of the Tutte polynomial!!

Rank Generating Function The rank generating function of the Tutte polynomial is defined as where and are the sets of edges and vertices of the graph,, and is the rank of.