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Some results on graph spectra. (English) Zbl 1015.05051
Summary: This paper presents a variety of results on graph spectra. The number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix. We establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors. It is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement. We generalize to any eigenvalue a condition for $$0$$ and $$-1$$ to be eigenvalues of a graph and its complement, respectively. Finally, we generalize to non-simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue.

MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
main eigenvalues
Full Text:
References:
  Bell, F.K.; Rowlinson, P., Certain graphs without zero as an eigenvalue, Math. japonica, 48, 961-967, (1993) · Zbl 0792.05094  Cvetković, D.M., Graphs and their spectra, Univ. beograd, publ. elektrotehn fak., ser. mat. fiz., 354-356, 1-50, (1971) · Zbl 0238.05102  Cvetković, D.M., The main part of the spectrum, divisors and switching of graphs, Publ. inst. math. (beograd), 23, 37, 31-38, (1978) · Zbl 0423.05028  Cvetković, D.; Doob, M., Developments in the theory of graph spectra, Linear and multilinear algebra, 18, 153-181, (1985) · Zbl 0615.05039  Cvetković, D.; Fowler, P.W., A group-theoretical bound for the number of main eigenvalues of a graph, J. chem. inf. comput. sci., 39, 638-641, (1999)  Cvetković, D.; Rowlinson, P.; Simić, S., ()  Harary, F.; Schwenk, A.J., The spectral approach to determining the number of walks in a graph, Pacific J. math., 80, 443-449, (1979) · Zbl 0417.05032  Li, Q.; Feng, K.Q., On the largest eigenvalue of a graph, Acta math. appl. sinica, 2, 167-175, (1979)  Mukherjee, A.K.; Datta, K.K., Two new graph-theoretical methods for the generation of eigenvectors of chemical graphs, Proc. Indian acad. sci. chem. sci., 101, 499-517, (1989)  Powers, D.L.; Sulaiman, M.M., The walk partition and colorations of graphs, Linear algebra appl., 48, 145-159, (1982) · Zbl 0501.05044
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