Upper bounds for the sum of Laplacian eigenvalues of a graph and Brouwer’s conjecture
Hilal A. Ganie, S. Pirzada, Rezwan Ul Shaban, X. Li
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Source: Crossref
Published: Apr 1, 2019
DOI: 10.1142/s1793830919500289
Open original source ↗Source abstract
Consider a simple graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] having Laplacian eigenvalues [Formula: see text]. Let [Formula: see text] be the sum of [Formula: see text] largest Laplacian eigenvalues of [Formula: see text]. Brouwer conjectured that [Formula: see text] for all [Formula: see text]. We obtain an upper bound for [Formula: see text] in terms of the clique number [Formula: see text], the number of vertices [Formula: see text] and the non-negative integers [Formula: see text] associated to the structure of the graph [Formula: see text]. We show that the Brouwer’s conjecture holds true for some new families of graphs. We use the same technique to prove that the Brouwer’s conjecture is true for a subclass of split graphs (It is already known that Brouwer’s conjecture holds for split graphs).
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