The paper proves the localized bound q(G) ≤ 2 ∑_v (1 − 1/c(v)) on the signless Laplacian spectral radius, where c(v) is the size of the largest clique containing vertex v, with equality cases for certain complete multipartite graphs.
On the notion of balance of a signed graph.Michigan Math
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Localized Tur\'{a}n-type inequalities for $Q$-index
The paper proves the localized bound q(G) ≤ 2 ∑_v (1 − 1/c(v)) on the signless Laplacian spectral radius, where c(v) is the size of the largest clique containing vertex v, with equality cases for certain complete multipartite graphs.