ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 108509, 21 pages
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. Trinajstić
Copyright © 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.
- I. Sciriha and I. Gutman, “Nut graphs: maximally extending cores,” Utilitas Mathematica, vol. 54, pp. 257–272, 1998.
- F. R. Gantmacher II, The Theory of Matrices, Chelsea, New York, NY, USA, 1960.
- V. Chvátal and P. L. Hammer, “Aggregation of inequalities in integer programming,” in Studies in Integer Programming (Proc. Workshop, Bonn, 1975), P. L. Hammer, E. L. Johnson, B. H. Korte, et al., Eds., Annals of Discrete Mathematics, 1, pp. 145–162, North-Holland, Amsterdam, The Netherlands, 1977.
- P. B. Henderson and Y. Zalcstein, “A graph-theoretic characterization of the PV class of synchronizing primitives,” SIAM Journal on Computing, vol. 6, no. 1, pp. 88–108, 1977.
- M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, vol. 57 of Annals of Discrete Mathematics, Elsevier Science, Amsterdam, The Netherlands, 2nd edition, 2004.
- N. V. R. Mahadev and U. N. Peled, Threshold Graphs and Related Topics, vol. 56 of Annals of Discrete Mathematics, North-Holland, Amsterdam, The Netherlands, 1995.
- R. Merris, Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, NY, USA, 2001.
- A. Brandstädt, V. B. Le, and J. P. Spinrad, Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1999.
- H. A. Jung, “On a class of posets and the corresponding comparability graphs,” Journal of Combinatorial Theory. Series B, vol. 24, no. 2, pp. 125–133, 1978.
- H. Lerchs, “On cliques and kernels,” Tech. Rep., Department of Computer Science, University of Toronto, 1971.
- D. Seinsche, “On a property of the class of n-colorable graphs,” Journal of Combinatorial Theory. Series B, vol. 16, pp. 191–193, 1974.
- D. P. Sumner, “Dacey graphs,” Journal of the Australian Mathematical Society, vol. 18, pp. 492–502, 1974.
- T. Biyikoglu, S. K. Simi, and Z. Stanić, “Some notes on spectra of cographs,” Ars Combinatoria, vol. 100, pp. 421–434, 2011.
- D. G. Corneil, Y. Perl, and L. K. Stewart, “A linear recognition algorithm for cographs,” SIAM Journal on Computing, vol. 14, no. 4, pp. 926–934, 1985.
- E. Ruch and I. Gutman, “The branching extent of graphs,” Journal of Combinatorics, Information & System Sciences, vol. 4, no. 4, pp. 285–295, 1979.
- R. A. Brualdi and A. J. Hoffman, “On the spectral radius of (0, 1)-matrices,” Linear Algebra and Its Applications, vol. 65, pp. 133–146, 1985.
- R. A. Brualdi and E. S. Solheid, “On the spectral radius of connected graphs,” Institut Mathématique. Publications. Nouvelle Série, vol. 39(53), pp. 45–54, 1986.
- Z. Stanić, “On nested split graphs whose second largest eigenvalue is less than 1,” Linear Algebra and Its Applications, vol. 430, no. 8-9, pp. 2200–2211, 2009.
- I. Sciriha, “A characterization of singular graphs,” Electronic Journal of Linear Algebra, vol. 16, pp. 451–462, 2007.
- I. Sciriha, “On the rank of graphs,” in Combinatorics, Graph Theory, and Algorithms, Y. Alavi, D. R. Lick, and A. Schwenk, Eds., vol. 2, pp. 769–778, New Issues Press, Kalamazoo, Mich, USA, 1999.
- I. Sciriha, “On the construction of graphs of nullity one,” Discrete Mathematics, vol. 181, no. 1–3, pp. 193–211, 1998.
- I. Sciriha, S. Fiorini, and J. Lauri, “Minimal basis for a vector space with an application to singular graphs,” Graph Theory Notes of New York, vol. 31, pp. 21–24, 1996.
- I. Sciriha, “Maximal core size in singular graphs,” Ars Mathematica Contemporanea, vol. 2, no. 2, pp. 217–229, 2009.
- G. F. Royle, “The rank of a cograph,” The Electronic Journal of Combinatorics, vol. 10, no. 1, 2003.
- D. Cvetković and M. Petrić, “A table of connected graphs on six vertices,” Discrete Mathematics, vol. 50, no. 1, pp. 37–49, 1984.
- E. M. Hagos, “Some results on graph spectra,” Linear Algebra and Its Applications, vol. 356, pp. 103–111, 2002.
- I. Sciriha and D. M. Cardoso, “Necessary and sufficient conditions for a hamiltonian graph,” Journal of Combinatorial Mathematics and Combinatorial Computing. In press.
- D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Johann Ambrosius Barth, Heidelberg, Germany, 3rd edition, 1995.
- R. Merris, “Antiregular graphs are universal for trees,” Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika, vol. 14, pp. 1–3, 2003.
- R. B. Bapat, Graphs and Matrice, Hindustan Book Agency, New Delhi, India, 2011.
- H. Bai, “The Grone-Merris conjecture,” Transactions of the American Mathematical Society, vol. 363, no. 8, pp. 4463–4474, 2011.