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Journal of the Ramanujan Mathematical Society

Volume 36, Issue 1, March 2021  pp. 13–21.

On Laplacian eigenvalues of graphs and Brouwer's conjecture

Authors:  Hilal A. Ganie, S. Pirzada, Bilal A. Rather and Rezwan Ul Shaban
Author institution:Department of School Education, JK Govt. Kashmir, India

Summary:  Let μ{1}, μ{2}, …, μ{n-1}, μ{n} = 0 be the Laplacian eigenvalues of a simple graph G of order n and size m. Let S{k} (G) = ∑{i = 1}{k} μ{i} be the sum of k largest Laplacian eigenvalues of G. In 2010, Brouwer conjectured that S{k} (G) ≤ m + (k+1 2), for all k = 1, 2, …, n. This conjecture has attracted much attention of the researchers because of the importance of the parameter S{k} (G) in spectral graph theory and it has been shown that it holds for various families of graphs. In this paper, we put conditions on the number of edges m in terms of the order of the graph n and the positive integers p and q to guarantee the truth of Brouwer's conjecture. This generalizes the result of Chen [X. Chen, Improved results on Brouwer's conjecture for sum of the Laplacian eigenvalues of a graph, Linear Algebra Appl. 557 (2018) 327–338]. Under certain conditions, we show that Brouwer's conjecture is true for biregular graphs and split graphs with a cycle C{t} fused at a vertex of the clique. As a consequence, this gives new families of graphs for which the spectral threshold dominance property holds for some values of k.

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