#### Title

The Third Step Seymour Vertex

#### Format

Oral Presentation

#### Faculty Mentor Name

Larry Langley

#### Faculty Mentor Department

Mathematics

#### Abstract/Artist Statement

In a social network, does there exist a person with at least as many friends of friends as they have friends? In 1990, Paul Seymour conjectured that this is always true, no matter how complex the network. In more mathematical language, Seymour’s Second Neighborhood Conjecture states that there exists at least one vertex in a directed graph G such that the second out neighborhood|N++(x)|, the set of all vertices distance 2 from vertex, is at least as large as its first out-neighborhood, denoted by|N+(x)|, that is that|N++(x)|≥|N+(x)|.We extend this conjecture to the next step, suggesting that there exists a person in a social network that has as many friends of friends of friends as they have friends. We conjecture that this property will hold for the third out-neighborhood, that is that|N+++(x)| ≥ |N+(x)|.Specifically, we look at complete bipartite directed graphs and begin laying foundation to show that this property holds for all Km,n.

#### Location

University of the Pacific, 3601 Pacific Ave., Stockton, CA 95211

#### Start Date

24-4-2021 11:30 AM

#### End Date

24-4-2021 11:45 AM

The Third Step Seymour Vertex

University of the Pacific, 3601 Pacific Ave., Stockton, CA 95211

In a social network, does there exist a person with at least as many friends of friends as they have friends? In 1990, Paul Seymour conjectured that this is always true, no matter how complex the network. In more mathematical language, Seymour’s Second Neighborhood Conjecture states that there exists at least one vertex in a directed graph G such that the second out neighborhood|N++(x)|, the set of all vertices distance 2 from vertex, is at least as large as its first out-neighborhood, denoted by|N+(x)|, that is that|N++(x)|≥|N+(x)|.We extend this conjecture to the next step, suggesting that there exists a person in a social network that has as many friends of friends of friends as they have friends. We conjecture that this property will hold for the third out-neighborhood, that is that|N+++(x)| ≥ |N+(x)|.Specifically, we look at complete bipartite directed graphs and begin laying foundation to show that this property holds for all Km,n.