For a simple graph G with n vertices and m edges, let D(G)=\documentclass[12pt]{minimal}
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\begin{document}$$D(G)=$$\end{document} diag(d1,d2,⋯,dn)\documentclass[12pt]{minimal}
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\begin{document}$$(d_1, d_2, \dots , d_n)$$\end{document} be its diagonal matrix, where di=deg(vi),\documentclass[12pt]{minimal}
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\begin{document}$$d_i=\deg (v_i),$$\end{document} for all i=1,2,⋯,n\documentclass[12pt]{minimal}
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\begin{document}$$i=1,2,\dots ,n$$\end{document} and A(G) be its adjacency matrix. The matrix Q(G)=D(G)+A(G)\documentclass[12pt]{minimal}
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\begin{document}$$Q(G)=D(G)+A(G)$$\end{document} is called the signless Laplacian matrix of G. If q1,q2,⋯,qn\documentclass[12pt]{minimal}
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\begin{document}$$q_1,q_2,\dots ,q_n$$\end{document} are the signless Laplacian eigenvalues of Q(G) arranged in a non-increasing order, let Sk+(G)=∑i=1kqi\documentclass[12pt]{minimal}
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\begin{document}$$S^{+}_{k}(G)=\sum _{i=1}^{k}q_i$$\end{document} be the sum of the k largest signless Laplacian eigenvalues of G. As the signless Laplacian matrix Q(G) is a positive semi-definite real symmetric matrix, so the spectral invariant Sk+(G)\documentclass[12pt]{minimal}
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\begin{document}$$S^{+}_{k}(G)$$\end{document} actually represents the Ky Fan k-norm of the matrix Q(G). Ashraf et al. (Linear Algebra Appl 438:4539–4546, 2013) conjectured that [inline-graphic not available: see fulltext], for all k=1,2,⋯,n\documentclass[12pt]{minimal}
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\begin{document}$$k=1,2,\dots ,n$$\end{document}. In this paper, we obtain upper bounds to Sk+(G)\documentclass[12pt]{minimal}
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\begin{document}$$S^{+}_{k}(G)$$\end{document} for some infinite families of graphs. Those structural results and tools are applied to show that the conjecture holds for many classes of graphs, and in particular for graphs with a given clique number.
