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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 108509, 21 pages
http://dx.doi.org/10.5402/2011/108509
Research Article

On the Spectrum of Threshold Graphs

Mathematics Department, Faculty of Science, University of Malta, Msida MSD 2080, Malta

Received 29 September 2011; Accepted 3 November 2011

Academic Editors: F. Ergun, P. Feinsilver, Y. Hou, S. Y. Song, and N. I. Trinajstiฤ‡

Copyright ยฉ 2011 Irene Sciriha and Stephanie Farrugia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The antiregular connected graph on ๐‘Ÿ vertices is defined as the connected graph whose vertex degrees take the values of ๐‘Ÿโˆ’1 distinct positive integers. We explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number ๐‘Ÿ of parts. Structural and combinatorial properties can be deduced for related classes of graphs and in particular for the minimal configurations in the class of singular graphs.