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

Event

Daniel Roberts

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

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.

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Apr 21st, 2:00 PM Apr 21st, 3:00 PM

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.