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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.
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