Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

Direct grafting poly(methyl methacrylate) from TiO2 nanoparticles via Cu2+‐amine redox‐initiated radical polymerization: An advantage of monocenter initiation

“…Different from Cu 31 /OH 2 and Ce 41 /H 1 , air-regenerative Cu 21 can work under mild conditions with amines to initiate radical polymerization [41][42][43][44][45][46]. Recently, we reported continuous redox-initiated radical polymerization from lowmolecular weight (MW), high-MW, or surface-confined amines based on Cu 1 -Cu 21 regeneration in the presence of O 2 [47][48][49]. Beyond the preliminary results reported earlier [50], in this report, we examined the CuSO 4 -catalyzed redox-initiated radical polymerization of MMA from amine-functionalized TiO 2 nanoparticles (TiO 2 -NH 2 nanoparticles) in the presence of O 2 .…”

Characterization of tricyclic graphs with exactly two $Q$-main eigenvalues

“…[6,7,8,9,10,14]). Motivated by these works, Chen and Huang [13] obtained the following results on graphs with Q-main eigenvalues: Theorem 1.1 ( [13]). G contains just one Q-main eigenvalue if and only if G is regular.…”

“…G contains just one Q-main eigenvalue if and only if G is regular. Theorem 1.2 ( [13]). Let G be a graph with signless Laplacian Q.…”

“…For convenience, let G a,b be the set of the connected graphs that satisfying (1.1), where (a, b) is the parameters of G ∈ G a,b . We know that a > 0 and b ≤ 0 from [13].…”

“…In particular, Chen and Huang [13] characterized all trees, unicylic graphs and bicyclic graphs with exactly two main Q-eigenvalues, respectively. As a continuance of it, in this paper we consider Q-main eigenvalues of tricyclic graphs.…”