# The Spectrum Problem for Digraphs of Order 4 and Size 6

## Submission Type

Event

## Faculty Advisor

Daniel Roberts

## Expected Graduation Date

2018

## Location

Center for Natural Sciences, Illinois Wesleyan University

## Start Date

4-21-2018 2:00 PM

## End Date

4-21-2018 3:00 PM

## Disciplines

Education

## Abstract

In graph theory, graph decomposition is a typical problem. An H-decomposition of G is also called a -design, where G and H are graphs. A complete digraph, ��^{*}_{��}, can be obtained by adding one repeated edge to each edge of ��^{*}_{��}, which denotes complete graph with n vertices. The spectrum for a digraph H is the set of all n for which a (��^{*}_{��, }H)-design exists. Now, let D be any directed digraph obtained by orienting the edges of a paw graph with two double edges. The *paw graph* consists of a triangle with a pendant edge attached to one of the three vertices. From the spectra of *paw*, we found 18 possibilities of such D. Our goal is to settle the spectrum for such D. For 12 of the 18 possibilities, we establish necessary and sufficient conditions on n for the existence of a (��^{*}_{��}, D)-design. Partial results are given for the remaining 6 possibilities of D.

The Spectrum Problem for Digraphs of Order 4 and Size 6

Center for Natural Sciences, Illinois Wesleyan University

In graph theory, graph decomposition is a typical problem. An H-decomposition of G is also called a -design, where G and H are graphs. A complete digraph, ��^{*}_{��}, can be obtained by adding one repeated edge to each edge of ��^{*}_{��}, which denotes complete graph with n vertices. The spectrum for a digraph H is the set of all n for which a (��^{*}_{��, }H)-design exists. Now, let D be any directed digraph obtained by orienting the edges of a paw graph with two double edges. The *paw graph* consists of a triangle with a pendant edge attached to one of the three vertices. From the spectra of *paw*, we found 18 possibilities of such D. Our goal is to settle the spectrum for such D. For 12 of the 18 possibilities, we establish necessary and sufficient conditions on n for the existence of a (��^{*}_{��}, D)-design. Partial results are given for the remaining 6 possibilities of D.