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2000, iss. 4, pp. 51-58
On eigenvalues and main eigenvalues of a graph
Kragujevac

emaillepovic@uis0.uis.kg.ac.yu
Keywords: graph; eigenvalue; main eigenvalue
Abstract
Let G be a simple graph of order n and let λ1 ≥ λ 2 ≥ ··· ≥ n and λ1 ≥ λ2 ≥ ··· ≥ λn be its eigenvalues with respect to the ordinary adjacency matrix A = A(G) and the Seidel adjacency matrix A*=A*(G), respectively. Using the Courant-Weyl inequalities we prove that λ n+1−i Є [−λ i−1, λ i+1−1] and λ n*+1−i Є [−2 λ i−1,−2 λ i+1−1] for i = 1, 2,..., n−1, where λ i are the eigenvalues of its complement G. Besides, if G and H are two switching equivalent graphs then we find λ i(G) Є [λ i+1(H), λ i−1(H)] for i = 2,3,. .., n − 1. Next, let μ1, μ2,..., μk and π1, π2,..., π k denote the main eigenvalues of the graph G and the complementary graph G, respectively. In this paper we also prove: Σ k i=1 (μi + πi) = n - k.
References
Cvetković, D.M., Doob, M., Sachs, H. (1995) Spectra of graphs: Theory and application. Heidelberg-Leipzig: Barth Verlag
Cvetković, D.M., Rowlinson, P., Simić, S.K. (1997) Eigenspaces of graphs. in: Encyclopedia of Mathematics and its Applications, Cambridge, itd: Cambridge University Press, vol. 66
Cvetković, D.M. (1971) Graphs and their spectra. Publ. Elektrotehničkog fakulteta - serija: matematika i fizika, br. 356, 1-50
Cvetković, D.M. (1978) The main part of the spectrum, divisors and switching of graphs. Publications de l'Institut mathematique, 23(37), 31-38
Cvetković, D.M. (1982) On graphs whose second largest eigenvalue does not exceed $1$. Publications de l'Institut mathematique, 31(45), 15-20
Lepović, M. (2000) On spectral complementary graphs. Journal of Mathematics, Novi Sad, vol. 30, br. 3, str. 83-91
Lepović, M.V. (1998) On formal products and the Seidel spectrum of graphs. Publications de l'Institut mathematique, 63(77), 37-46
 

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article language: English
document type: Paper
DOI: 10.5937/MatMor0004051L
published in SCIndeks: 05/05/2008

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