A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3}, satisfying the condition that every vertex u for which f(u)=1 adjacent to at least one assigned 2 or 3, and with f(u)=0 3 two vertices 2. The weight of f equals sum w(f)=?v?Vf(v). minimum G called domination number ?dR(G) G. We obtain tight bounds in some cases closed expressions generalized Petersen graphs P(ck,k). In shor...

We obtain $$L^p$$ estimates for Toeplitz operators on the generalized Hartogs triangles $$\mathbb {H}_\gamma = \{(z_1,z_2) \in \mathbb {C}^2\,{:}\, |z_1|^\gamma \!< |z_2|<1\}$$ two classes of positive radial symbols, one a power distance to origin, and other boundary.

New results on singleton rainbow domination numbers of generalized Petersen graphs P(ck,k) are given. Exact values established for some infinite families, and lower upper bounds with small gaps given in all other cases.