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.