Laplacian spectral characterization of setosa graphs

Document Type : Research Paper

Author

Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran.

Abstract

A setosa graph $SG(e,f,g,h,d;b_1,b_2,\ldots,b_s)$ is a graph consisting of five cycles and $s(\geq 1)$ paths $P_{b_1+1}, P_{b_2+1},\ldots,P_{b_s+1}$ intersecting in a single vertex that all meet in one vertex, where $b_i\geq1$ (for $i=1,\ldots,s$) and $e,f,g,h,d\geq 3$ denote the length of the cycles $C_e$, $C_f$, $C_g$, $C_h$ and $C_d$, respectively. Two graphs $G$ and $H$ are $L$-cospectral if they have the same Laplacian spectrum. A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS, for short) if every graph with the same Laplacian spectrum is isomorphic to $G$. In this paper we prove that if $H$ is a $L$-cospectral graph with a setosa graph $G$, then $H$ is also a setosa graph and the degree sequence of $G$ and $H$ are the same. We conjecture that all setosa graphs are DLS.

Keywords

References

[1] A. Z. Abdian, Bell graphs are determined by their Laplacian spectra, Kragujevac J. Math., 47 No. 2 (2023) 203-211.
[2] A. Z. Abdian, A. R. Ashrafi and M. Brunetti, Signless Laplacian spectral characterization of roses, Kuwait J. Sci., 47 No. 4 (2021) 12-18.
[3] A.Z. Abdian, L. W. Beineke, K. Thulasiraman, R. Tayebi Khorami and M. R. Oboudi, The spectral determination of the connected multicone graphs KwrrCs, AKCE Int. J. Graphs Comb., 18 (2021) 47-52.
[4] A. Z. Abdian, K. Thulasiraman and K. Zhao, The spectral characterization of the connected multicone graphs, AKCE Int. J. Graphs Comb., 17 No. 1 (2020) 606-613.
[5] A. Z. Abdian, K. Thulasiraman and K. Zhao, The spectral determination of the multicone graphs KwrC with respect to their signless Laplacian spectra, J. Algebr. Syst., 7 No. 2 (2019) 131-141.
[6] W. N. Anderson and T. D. Morley, Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebra, 18 (1985) 141-145.
[7] D. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra, London Math-ematical Society Student Texts 75, Cambridge University Press, Cambridge, 2010.
[8] C. Godsil and G. Royle, Algebraic Graph Theory, Springer, New York, 2001.
[9] R. Grone and R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math., 7 No. 2 (1994) 221-229.
[10] J. S. Li and Y. L. Pan, A note on the second largest eigenvalue of the Laplacian matrix of a graph, Linear Multilinear Algebra, 48 (2000) 117-121.
[11] F. J. Liu and Q. X. Huang, Laplacian spectral characterization of 3-rose graphs, Linear Algebra Appl., 439 (2013) 2914-2920.
[12] R. Merris, A note on the Laplacian graph eigenvalues, Linear Algebra Appl., 285 (1998) 33-35.
[13] M. R. Oboudi and A. Z. Abdian, Peacock graphs are determined by their Laplacian spectra, Iran. J. Sci. Technol. Trans. A Sci., 44 (2020) 787-790.
[14] M. R. Oboudi, A. Z. Abdian, A. R. Ashrafi and L. W. Beineke, Laplacian spectral determination of path-friendship graphs, AKCE Int. J. Graphs Comb., 18 (2021) 33-38.
[15] 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., 356 (2002) 113-121.
[16] G. R. Omidi, On a signless Laplacian spectral characterization of T-shape trees, Linear Algebra Appl., 431 (2009) 1607-1615.
[17] E. R. van Dam, W. H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl., 373 (2003) 241-272.
[18] F. Wen, Q. Huang, X. Huang, F. Liu, The spectral characterization of wind-wheel graphs, Indian J. Pure Appl. Math., 46 (2015) 613-631.
Algebraic Structures and Their Applications
Volume 10, Issue 1
February 2023
Pages 39-46

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