Analogue of Brauer’s conjecture for the signless Laplacian of cographs
https://doi.org/10.29235/1561-2430-2020-56-3-310-317
Abstract
In this paper, we consider the class of cographs and its subclasses, namely, threshold graphs and anti-regular graphs. In 2011 H. Bai confirmed the Grone – Merris conjecture about the sum of the first k eigenvalues of the Laplacian of an arbitrary graph. As a variation of the Grone – Merris conjecture, A. Brouwer put forward his conjecture about an upper bound for this sum. Although the latter conjecture was confirmed for many graph classes, however, it remains open. By analogy to Brouwer’s conjecture, in 2013 F. Ashraf et al. put forward a conjecture about the sum of k eigenvalues of the signless Laplacian, which was also confirmed for some graph classes but remains open. In this paper, an analogue of the Brouwer’s conjecture is confirmed for the graph classes under our consideration for the eigenvalues of their signless Laplacian for some natural k which does not exceed the order of the considered graphs.
Keywords
cograph,
threshold graph,
anti-regular graph,
adjacency matrix,
Laplacian,
signless Laplacian,
spectra of matrix
Supporting agencies: This work was funded by the Institute of Mathematics of the National Academy of Sciences of the Republic of Belarus within the framework of “Convergence” State Program for Fundamental Research and the Belarusian Republican Foundation for Fundamental Research (project no. Ф20УКА-005).
About the Author
V. I. Benediktovich
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Vladimir I. Benediktovich – Ph. D. (Physics and Mathematics), Leading Researcher
Surganov Str., 11, 220072, Minsk
References
1. Bai H. Grone-Merris Conjecture. Transactions of the American Mathematical Society, 2011, vol. 363, no. 8, pp. 4463–4474. https://doi.org/10.1090/s0002-9947-2011-05393-6
2. Grone R., Merris R. The Laplacian Spectrum of a Graph II. SIAM Journal on Discrete Mathematics, 1994, vol. 7, no. 2, pp. 221–229. https://doi.org/10.1137/s0895480191222653
3. Brouwer A. E., Haemers W. H. Spectra of Graphs. New York, Springer, 2012. 250 p. https://doi.org/10.1007/978-1-4614-1939-6
4. Ashraf F., Omidi G. R., Tayfeh-Rezaie B. On the sum of signless Laplacian eigenvalues of a graph. Linear Algebra and its Applications, 2013, vol. 438, no. 11, pp. 4539–4546. https://doi.org/10.1016/j.laa.2013.01.023
5. Chvátal V., Hammer P. L. Set-packing and threshold graphs. Technical Report Rep., CORR 73–21. Ontario, 1973.
6. Yang, J., You L. On a conjecture for the signless Laplacian eigenvalues. Linear Algebra and its Applications, 2014, vol. 446, pp. 115–132. https://doi.org/10.1016/j.laa.2013.12.032
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