#### Title

On Copoint Graphs

#### Faculty Sponsor

Jon Beagley

#### College

Arts and Sciences

#### Department/Program

VERUM

#### ORCID Identifier(s)

0000-0003-4666-9538, 0000-0002-6343-4362, 0000-0001-5914-2140

#### Presentation Type

Poster Presentation

#### Symposium Date

Summer 7-29-2019

#### Abstract

A convex geometry is a discrete abstraction of convexity defined by a meet-distributive lattice on a finite set. In particular, we study a graph formed from the copoints of a convex geometry. A graph that can be realized in this way from some convex geometry is called a copoint graph. We demonstrate existence and non-existence for several infinite families of graphs as copoint graphs. We show that the graph join of a copoint graph and a non-copoint graph is not a copoint graph. Further, we provide a construction to show that the complement of a copoint graph need not be a copoint graph. We conclude that not all trees are copoint graphs and argue that the Hasse diagram of a convex geometry has a `rhomboidal' structure if and only if its copoint graph is a tree.

#### Recommended Citation

Albert, Michael; Franchere, Evan; and Zomkyi, Tenzin, "On Copoint Graphs" (2019). *Summer Interdisciplinary Research Symposium*. 54.

https://scholar.valpo.edu/sires/54

## Biographical Information about Author(s)

Evan Franchere is a rising junior at Reed College majoring in mathematics. He became interested in discrete geometry when taking a topics in geometry course his sophomore year of college and as a result applied to Dr. Beagley's VERUM research project. Evan's current goal is to graduate college and eventually obtain a PhD in mathematics.

Michael Albert is a senior at Western Washington University majoring in mathematics. His interests are primarily analysis and topology. Michael will graduate in December 2019 and then plans to pursue a PhD in mathematics.

Tenzin Zomkyi is a rising senior at York College (CUNY) majoring in mathematics and minoring in biology. She is interested in bio-math specifically in cellular biology, but she might consider other fields in math if that interest her. Her future goal is to pursue a PhD in mathematics.