Properties of the characteristic polynomials and spectrum of Pn and Cn


  • Essam El Seidy
  • Salah Eldin Hussein
  • Atef Mohamed Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt





Laplacian Matrix, Signless Laplacian Matrix, Normalized Laplacian Matrix, Seidel Adjacency Matrix, Spectral.


We consider a finite undirected and connected simple graph  with vertex set  and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.


[1] Ayoobi,F., Omidi,G. R. and Tayfeh-Rezaie,B. “A note on graphs whosesignlessLaplacian has three distinct eigenvaluesâ€, Lin. Multilin. Alg. 59(2011) 701–706. (p. 220)

[2] Biggs, N.L. “Algebraic Graph Theoryâ€, Cambridge University Press, 2nd edition,Cambridge, 1993.

[3] Bondarenko, A. V. and Radchenko,D. V. “On a family of strongly regulargraphs with λ = 1â€, arXiv:1201.0383, Feb. 2012. (p. 132).

[4] Boulet, R.“Disjoint unions of complete graphs characterized by their Laplacian spectrumâ€, Electr. J. Lin. Alg. 18 (2009) 773–783. (p. 204).

[5] Cvetković, D., Doob, M. and H. Sachs, H. “Spectra of Graphs, Theory andApplicationsâ€, Academic Press, 1980.

[6] Cvetkovic, D., Rowlinson, P., and Simic, S. (2010) “An Introduction to the Theory of Graph Spectra Cambridge U.P.â€, New York.

[7] Das, K.C. (2004) “The Laplacian spectrum of a graphâ€,Comput Math.Appl.48,715724.

[8] Davis,P.J. “Circulant Matricesâ€, AMS Chelsea Publishing, 1994.

[9] El seedy, E., Hussein, S. and AboElkher, A. “Spectra ofsome simple graphsâ€,Math. Theory and Mod. 5 (2015) 115-121.

[10] Fiedler, M. “Algebraic connectivity of graphsâ€, Czech. Math. J., 1973; 23:298-305.

[11] Godsil C. and Royle, G. “Algebraic Graph Theoryâ€, Springer Verlag, New York, 2001.

[12] Kaveh, A. “Structural Mechanics: Graph and Matrix Methodsâ€, 3rd edition, ResearchStudies Press, Somerset, UK, 2004.

[13] Kaveh, A. “Optimal Structural Analysisâ€, 2nd edition, John Wiley, UK, 2006.

[14] Kel’mans,A.K. “The number of trees in a graphâ€. I. Automat. iTelemeh. 26 (1965)2194–2204 (in Russian); transl. Automat. Remote Control 26 (1965) 2118–2129.

[15] Mohar, B. “The Laplacian spectrum of graphs, entitled Graph Theory, Combinatorics and Applicationsâ€, edit. Y. Alavi et al., Vol. 2, John Wiley, NY, 1991, pp 871-898.

[16] Pothen, A. Simon, H. and Liou, K.P. “Partitioning sparse matrices with eigenvectors of graphsâ€, SIAM J. Matrix Anal. Appl., 1990; 11:430-452.

[17] Spielman, D. (2004) “Spectral graph theory and its Applicationsâ€, lecture notes fall, onlinevia

[18] Topping, BHV and Sziveri, J. “Parallel subdomain generation methodâ€, Proc. CivilComp, Edinburgh, UK, 1995.

View Full Article: