Opuscula Math. 38, no. 1 (2018), 15-30

Opuscula Mathematica

The spectrum problem for digraphs of order 4 and size 5

Ryan C. Bunge
Steven DeShong
Saad I. El-Zanati
Alexander Fischer
Dan P. Roberts
Lawrence Teng

Abstract. The paw graph consists of a triangle with a pendant edge attached to one of the three vertices. We obtain a multigraph by adding exactly one repeated edge to the paw. Now, let \(D\) be a directed graph obtained by orientating the edges of that multigraph. For 12 of the 18 possibilities for \(D\), we establish necessary and sufficient conditions on \(n\) for the existence of a \((K^{*}_{n},D)\)-design. Partial results are given for the remaining 6 possibilities for \(D\).

Keywords: spectrum problem, digraph decompositions.

Mathematics Subject Classification: 05C20, 05C51.

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  • Ryan C. Bunge
  • Illinois State University, Normal, IL 61790-4520, USA
  • Steven DeShong
  • Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
  • Saad I. El-Zanati
  • Illinois State University, Normal, IL 61790-4520, USA
  • Alexander Fischer
  • Jacobs High School, Algonquin, IL 60102, USA
  • Dan P. Roberts
  • Illinois Wesleyan University, Bloomington, IL 61701, USA
  • Lawrence Teng
  • University of Michigan, Ann Arbor, MI 48109, USA
  • Communicated by Mariusz Meszka.
  • Received: 2016-01-05.
  • Revised: 2017-06-06.
  • Accepted: 2017-06-14.
  • Published online: 2017-11-13.
Opuscula Mathematica - cover

Cite this article as:
Ryan C. Bunge, Steven DeShong, Saad I. El-Zanati, Alexander Fischer, Dan P. Roberts, Lawrence Teng, The spectrum problem for digraphs of order 4 and size 5, Opuscula Math. 38, no. 1 (2018), 15-30, https://doi.org/10.7494/OpMath.2018.38.1.15

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