Maxima of the Q-index, Forbidden 4-ycle and 5-cycle
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Publication:5746853
DOI10.13001/1081-3810.1695zbMath1282.05166arXiv1308.1652OpenAlexW2963484794MaRDI QIDQ5746853
Maria Aguieras A. de Freitas, Laura Patuzzi, Vladimir Nikirofov
Publication date: 10 February 2014
Published in: The Electronic Journal of Linear Algebra (Search for Journal in Brave)
Abstract: This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an edge hanged to its center. It is shown that if G is a graph of order n, with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,k} be the join of a complete graph of order k and an independent set of order n-k. It is shown that if G is a graph of order n, with no 5-cycle, then q(G)<q(S_{n,2}), unless G=S_{n,k}. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q(G) of graphs with forbidden cycles.
Full work available at URL: https://arxiv.org/abs/1308.1652
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Extremal problems in graph theory (05C35) Paths and cycles (05C38) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50)
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