On the basis of an implicit iterative method for ill-posed operator equations, we introduce a relaxation factor w and a weighted factor μ, and obtain a stationary two-step implicit iterative method. The range of the factors which guarantee the convergence of iteration is explored. We also study the convergence properties and rates for both non-perturbed and perturbed equations. An implementable algorithm is presented by using Morozov discrepancy principle. The theoretical results show that the convergence rates of the new methods always lead to optimal convergent rates which are superior to those of the original one after choosing suitable relaxation and weighted factors. Numerical examples are also given, which coincide well with the theoretical results.
The spectral radius is an important parameter of a graph related to networks. A method for estimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of a Hamiltonian planar graph of order n≥4 is less than or equal to 2+3n-11 and the spectral radius of the outerplanar graph of order n≥6 is less than or equal to 22+n-5, which are improvements over previous results. A direction for further study is then suggested.
Let G be a graph of order n. G is called Hamiltonian if G has a spanning cycle. Denote N(v)={u∈V(G)｜uv∈E(G)}. Then by considering the degree and neighborhood union conditions of G, the paper gives a generalization of two theorems of Benhocine et al and Faudree et al. Let G be a 2\|connected graph of order n and α≤n2.If max{d(u),d(v)}≥n-12 or |N(u)∪N(v)|≥n-δ for every pair vertices u and v with d(u,v)= 2,then G is Hamiltonian with some exceptions.
A graph G is claw\|free if G has no induced subgraph isomorphic to K\-\{1,3\}. And a graph G is pancyclic if for every m, 3≤m≤|V(G)|, there is a cycle of length m. This paper considered neighbourhood union for any pair of nonadjacent vertices in claw\|free graph and obtained the following theorem: If G is a 2\|connected claw\|free graph of order n≥12 and |N(u)∪N(v)|+|N(u)∪N(w)|+|N(v)∪N(w)|≥2n-1 for any three pairwise nonadjacent vertices u,v, and w, then G is pancyclic.
