The Tutte polynomial

Gordon, Gary; McNulty, Jennifer

doi:10.1017/cbo9781139049443.010

Cited by 3 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some of these interpretations are closely related to matroidal or graphical properties of β. This lends support to our view that the Tutte and characteristic polynomials studied in [11,14,15,16,17,18,19] are (in some sense) also the 'right' generalizations to greedoids.…”

Section: Introductionsupporting

confidence: 69%

A $\beta$ Invariant for Greedoids and Antimatroids

Gordon

1997

Electron. J. Combin.

View full text Add to dashboard Cite

We extend Crapo's $\beta $ invariant from matroids to greedoids, concentrating especially on antimatroids. Several familiar expansions for $\beta (G)$ have greedoid analogs. We give combinatorial interpretations for $\beta (G)$ for simplicial shelling antimatroids associated with chordal graphs. When $G$ is this antimatroid and $b(G)$ is the number of blocks of the chordal graph $G$, we prove $\beta (G)=1-b(G)$.

show abstract

Section: Introductionsupporting

confidence: 69%

A $\beta$ Invariant for Greedoids and Antimatroids

Gordon

1997

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…There has been a sustained program organized to extend the Tutte polynomial to nonmatroidal structures; see [6,11,12,13,14] for a sample of this work. The probabilistic approach taken here should be applicable to many of the combinatorial structures considered here.…”

Section: Applications To Other Antimatroids and Greedoidsmentioning

confidence: 99%

When bad things happen to good trees*

Aivaliotis

Gordon

Graveman

2001

Journal of Graph Theory

View full text Add to dashboard Cite

When the edges in a tree or rooted tree fail with a certain ®xed probability, the (greedoid) rank may drop. We compute the expected rank as a polynomial in p and as a real number under the assumption of uniform distribution. We obtain several different expressions for this expected rank polynomial for both trees and rooted trees, one of which is especially simple in each case. We also prove two extremal theorems that determine both the largest and smallest values for the expected rank of a (rooted or unrooted) tree, and precisely when these extreme bounds are achieved. We conclude with directions for further study. ß

show abstract

“…As usual, a tree is an acyclic connected graph. A tree having at most one vertex of degree ≥ 3 is called a spider, [8], or an aster, [5].…”

Section: Introductionmentioning

confidence: 99%