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#### Title of Abstract

On Interval Divisor Graphs

10

#### College

College of Arts and Sciences

Mathematics

Dr. Arran Hamm

#### Abstract

A graph is a collection of vertices along with an edge relation, which is a rule by which vertices are joined by an edge. A divisor graph is a graph whose vertices are labeled by whole numbers and whose edge relation is given by divisibility (i.e., i and j are joined by an edge if i divides evenly into j or vice versa). Our work has focused on interval divisor graphs, for which the number labels are consecutive whole numbers. Specifically, one can easily observe that if the label “1” is present, the graph will be connected (since 1 divides evenly into all numbers). If, however, 1 is not in the given interval, then the interval divisor graph will not be connected. We give a criterion to check if two numbers are in the same connected component in a general interval divisor graph. We also examine the number of edges in an interval divisor graph by using a bit of Calculus II. We will finish by giving some partial results on finding the matching number of interval divisor graphs, where the matching number is the largest number of nonintersecting edges in a graph.

#### Grant Support?

Supported by a grant from the Winthrop University Research Council

#### Start Date

22-4-2016 12:00 PM

#### End Date

22-4-2016 2:00 PM

#### Share

COinS

Apr 22nd, 12:00 PM Apr 22nd, 2:00 PM

On Interval Divisor Graphs

Rutledge

A graph is a collection of vertices along with an edge relation, which is a rule by which vertices are joined by an edge. A divisor graph is a graph whose vertices are labeled by whole numbers and whose edge relation is given by divisibility (i.e., i and j are joined by an edge if i divides evenly into j or vice versa). Our work has focused on interval divisor graphs, for which the number labels are consecutive whole numbers. Specifically, one can easily observe that if the label “1” is present, the graph will be connected (since 1 divides evenly into all numbers). If, however, 1 is not in the given interval, then the interval divisor graph will not be connected. We give a criterion to check if two numbers are in the same connected component in a general interval divisor graph. We also examine the number of edges in an interval divisor graph by using a bit of Calculus II. We will finish by giving some partial results on finding the matching number of interval divisor graphs, where the matching number is the largest number of nonintersecting edges in a graph.