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On nested split graphs whose second largest eigenvalue is less than 1. (English) Zbl 1194.05098
Summary: We determine all nested split graphs (i.e. graphs having no induced subgraphs equal to $$2K_{2},P_{4}$$ or $$C_{4}$$) having the second largest eigenvalue less than 1 and give some data and observations regarding obtained results.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text:
##### References:
 [1] T. Bıyıkoğlu, S.K. Simić, Z. Stanić, Some notes on spectra of cographs, Ars Combin., in press. · Zbl 1265.05349 [2] Corneil, D.G.; Perl, Y.; Stewart, L.K., A linear recognition algorithm for cographs, SIAM J. comput., 14, 926-934, (1985) · Zbl 0575.68065 [3] Cvetković, D., On graphs whose second largest eigenvalue does not exceed 1, Publ. inst. math., 31, 45, 15-20, (1982) · Zbl 0522.05044 [4] Cvetković, D.; Doob, M.; Sachs, H., Spectra of graphs – theory and application, (1995), Johann Ambrosius Barth Verlag Heidelberg-Leipzig · Zbl 0824.05046 [5] Cvetković, D.; Rowlinson, P.; Simić, S., Eigenspaces of graphs, (1997), Cambridge Univ. Press Cambridge · Zbl 0878.05057 [6] Cvetković, D.; Rowlinson, P.; Simić, S., Spectral generalizations of line graphs – on line graphs with least eigenvalue $$- 2$$, London math. soc., lecture notes series, vol. 314, (2004), Cambridge University Press [7] Guo, Shu-Guang, On bicyclic graphs whose second largest eigenvalue does not exceed 1, Linear algebra appl., 407, 201-210, (2005) · Zbl 1073.05043 [8] M. Petrović, Z. Radosavljević, Spectrally constrained graphs, Faculty of Science, Kragujevac, Yugoslavia, 2001. [9] G.F. Royle, The rank of cographs, Electron. J. Comb. 10 (1), Research paper N11:7 p., 2003. [10] Simić, S.K.; Li Marzi, E.M.; Belardo, F., Connected graphs of fixed order and size with maximal index: structural considerations, Le matematiche, LIX, 349-365, (2004) · Zbl 1195.05045 [11] Stanić, Z., On graphs whose second largest eigenvalue equals 1 - the star complement technique, Linear algebra appl., 420, 700-710, (2007) · Zbl 1106.05067 [12] Stanić, Z., On regular graphs and coronas whose second largest eigenvalue does not exceed 1, Linear and multilinear algebra, (2008) · Zbl 1207.05124 [13] Z. Stanić, S.K. Simić, On graphs with unicyclic star complement for 1 as the second largest eigenvalue, in: N. Bokan, M. Djorić, A.T. Fomenko, Z. Rakić, B. Wegner, J. Wess (Eds.), Proceedings of the Conference Contemporary Geometry and Related Topics, Belgrade, June 26-July 2, 2005, Faculty of Mathematics, Belgrade, 2006, pp. 475-484. · Zbl 1142.05054 [14] Xu, Guang-Hui, On unicyclic graphs whose second largest eigenvalue does not exceed 1, Discrete appl. math., 136, 117-124, (2004) · Zbl 1032.05085
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