Two graphs are said to be Q-cospectral if they have the same signless Laplacian spectrum. A graph is said to be DQS if there are no other nonisomorphic graphs Q-cospectral with it. A tree is called double starlike if it has exactly two vertices of degree greater than 2. Let Hn(p, q) with n ≥ 2, p ≥ q ≥ 2, denote the double starlike tree obtained by attaching p pendant vertices to one pendant vertex of the path Pn and q pendant vertices to the other pendant vertex of Pn. We prove that Hn(p, q) is DQS for n ≥ 2, p ≥ q ≥ 2.
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References
Ch. Bu and J. Zhou, “Signless Laplacian spectral characterization of the cones over some regular graphs,” Linear Algebra Appl., 436, No. 9, 3634–3641 (2012).
Ch. Bu, J. Zhou, and H. B. Li, “Spectral determination of some chemical graphs,” Filomat, 26, No. 6, 1123–1131 (2012).
Ch. Bu and J. Zhou, “Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra,” Linear Algebra Appl., 436, 143–151 (2012).
D. Cvetković, P. Rowlinson, and S. Simić, “An introduction to the theory of graph spectra,” London Math. Soc. Stud. Texts, 75 (2010).
K. Ch. Das, “On conjectures involving second largest signless Laplacian eigenvalue of graphs,” Linear Algebra Appl., 432, No. 11, 3018–3029 (2010).
Hs. H. Günthard and H. Primas, “Zusammenhang von Graphtheorie und Mo–Theotie von Molekeln mit Systemen konjugierter Bindungen,” Helv. Chim. Acta, 39, 1645–1653 (1956).
J. S. Li and X. D. Zhang, “On the Laplacian eigenvalues of a graph,” Linear Algebra Appl., 285, No. 1-3, 305–307 (1998).
M. Liu, B. Liu, and F. Wei, “Graphs determined by their (signless) Laplacian spectra,” Electron. J. Linear Algebra, 22, 112–124 (2011).
X. Liu, Y. Zhang, and P. Lu, “One special double starlike graph is determined by its Laplacian spectrum,” Appl. Math. Lett., 22, No. 4, 435–438 (2009).
P. Lu and X. Liu, “Laplacian spectral characterization of some double starlike trees,” J. Harbin Eng. Univ., 37, No. 2, 242–247 (2016); arXiv:1205.6027v2[math.CO].
G. R. Omidi, “On a signless Laplacian spectral characterization of T-shape trees,” Linear Algebra Appl., 431, No. 9, 1600–1615 (2009).
G. R. Omidi and E. Vatandoost, “Starlike trees with maximum degree 4 are determined by their signless Laplacian spectra,” Electron. J. Linear Algebra, 20, 274–290 (2010).
C. S. Oliveira, N. M. M. de Abreu, and S. Jurkiewilz, “The characteristic polynomial of the Laplacian of graphs in (a, b)-linear cases,” Linear Algebra Appl., 365, 113–121 (2002).
E. R. van Dam and W. H. Haemers, “Which graphs are determined by their spectrum?,” Linear Algebra Appl., 373, 241–272 (2003).
J. Wang and F. Belardo, “A note on the signless Laplacian eigenvalues of graphs,” Linear Algebra Appl., 435, No. 10, 2585–2590 (2011).
J. Zhou and Ch. Bu, “Spectral characterization of line graphs of starlike trees,” Linear Multilinear Algebra, 61, No. 8, 1041–1050 (2013).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 9, pp. 1274–1284, September, 2021. Ukrainian DOI: 10.37863/umzh.v73i9.634.
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Sharafdini, R., Abdian, A.Z. & Behmaram, A. Signless Laplacian Determination for a Family of Double Starlike Trees. Ukr Math J 73, 1478–1490 (2022). https://doi.org/10.1007/s11253-022-02006-4
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DOI: https://doi.org/10.1007/s11253-022-02006-4