Elizabeth McMahon

Convexity and the Beta Invariant

Abstract. We apply a generalization of Crapos β invariant to finite subsets of Rn. For a finite subset C of the plane, we prove β(C)=|int (C)|, where |int (C)| is the number of interior points of C, i.e., the number of points of C which are not on the boundary of the convex hull of C . This gives the following formula: ΣK free(-1)|K|-1|K|=|int(C)|.

A characteristic polynomial for rooted graphs and rooted digraphs

Abstract We consider the one-variable characteristic polynomial p ( G ; λ ) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coefficients and the degree of p ( G ; λ ). In particular, | p ( G ;0)| is the number of acyclic orientations of G, while the degree of p ( G ; λ ) gives the size of the minimum tree cover (every edge of G is adjacent to some edge of T), and the leading coefficient gives the number of such covers. Finally, we consider the class of rooted fans in detail; here p ( G ; λ ) shows cyclotomic behavior.

Chordal graphs and the characteristic polynomial

A characteristic polynomial was recently defined for greedoids, generalizing the notion for matroids. When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs. In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressible in terms of the clique decomposition of the graph. From it, one obtains an expression for the number of blocks in the graph in terms of clique sizes.

Moving faces to other places: Facet derangements

Abstract Derangements are a popular topic in combinatorics classes. We study a generalization to face derangements of the n-dimensional hypercube. These derangements can be classified as odd or even, depending on whether the underlying isometry is direct or indirect, providing a link to abstract algebra. We emphasize the interplay between the geometry, algebra, and combinatorics of these sequences, with lots of pretty pictures.

Partitions of AG(4,3) into maximal caps

Abstract In a geometry, a maximal cap is a collection of points of largest size no three of which are collinear. In A G ( 4 , 3 ) , maximal caps contain 20 points; the 81 points of A G ( 4 , 3 ) can be partitioned into 4 mutually disjoint maximal caps together with a single point P , where every pair of points that makes a line with P lies entirely inside one of those caps. The caps in a partition can be paired up so that both pairs are either in exactly one partition or they are both in two different partitions. This difference determines the two equivalence classes of partitions of A G ( 4 , 3 ) under the action by affine transformations.

Partitions of A G ( 4 , 3 ) into maximal caps

Abstract In a geometry, a maximal cap is a collection of points of largest size no three of which are collinear. In A G ( 4 , 3 ) , maximal caps contain 20 points; the 81 points of A G ( 4 , 3 ) can be partitioned into 4 mutually disjoint maximal caps together with a single point P , where every pair of points that makes a line with P lies entirely inside one of those caps. The caps in a partition can be paired up so that both pairs are either in exactly one partition or they are both in two different partitions. This difference determines the two equivalence classes of partitions of A G ( 4 , 3 ) under the action by affine transformations.

Partitions of AG(4,3)AG(4,3) into maximal caps

Abstract In a geometry, a maximal cap is a collection of points of largest size no three of which are collinear. In A G ( 4 , 3 ) , maximal caps contain 20 points; the 81 points of A G ( 4 , 3 ) can be partitioned into 4 mutually disjoint maximal caps together with a single point P , where every pair of points that makes a line with P lies entirely inside one of those caps. The caps in a partition can be paired up so that both pairs are either in exactly one partition or they are both in two different partitions. This difference determines the two equivalence classes of partitions of A G ( 4 , 3 ) under the action by affine transformations.