On graphs whose signless Laplacian index does not exceed 4.5

Abstract Let A G and D G be respectively the adjacency matrix and the degree matrix of a graph G . The signless Laplacian matrix of G is defined as Q G = D G + A G . The Q -spectrum of G is the set of the eigenvalues together with their multiplicities of Q G . The Q -index of G is the maximum eigenvalue of Q G . The possibilities for developing a spectral theory of graphs based on the signless Laplacian matrices were discussed by Cvetkovic et al. [D. Cvetkovic, P. Rowlinson, S.K. Simic, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155–171]. In the latter paper the authors determine the graphs whose Q -index is in the interval [ 0 , 4 ] . In this paper, we investigate some properties of Q -spectra of graphs, especially for the limit points of the Q -index. By using these results, we characterize respectively the structures of graphs whose the Q -index lies in the intervals ( 4 , 2 + 5 ] , ( 2 + 5 , ϵ + 2 ] and ( ϵ + 2 , 4.5 ] , where ϵ = 1 3 ( ( 54 - 6 33 ) 1 3 + ( 54 + 6 33 ) 1 3 ) ≈ 2.382975767 .

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