Connected graphs having maximum signless Laplacian spectral radius

Let $ρ(G)$ denote the signless Laplacian spectral radius of a graph G, and let Cn,m be the family of connected graphs of order n and size m. A graph is Q-maximal if it maximizes $ρ(G)$ over this family. Motivated by a question of Chang and Tam (2011), we study this extremal problem using scalar identities and coronal methods for threshold graphs. From the Perron equations, we derive the identity SG(λ)=Ta,b(λ) at $λ=ρ(G)$, where a and b are the total numbers of zero-vertices and one-vertices. This yields a one-point comparison principle: for a connected threshold graph H with the same block totals, SH(z)>Ta,b(z) at some z≥ n implies $ρ(H)>z$, allowing spectral comparison without estimating individual Perron coordinates. We prove that every Q-maximal threshold graph has character at most three, so its vertex-sequence contains at most three zero--one block pairs. We also show that the quasi-star graph is Q-maximal whenever n≥5 and n-1≤ m<(n²+10n-8)/8. For certain dense parameter pairs, we construct connected threshold graphs with signless Laplacian spectral radii strictly larger than those of both the quasi-star and the connected quasi-complete graph. Thus the corresponding two-candidate assertion for connected graphs fails in general.