Labelling wheels for minimum sum number

A simple undirected graph G is called a sum graph if there exists a labelling L of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label L(w) = L(u) +L(v). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The sum number oe(H) of a connected graph H is the least number r of isolated vertices K r such that G = H+K r is a sum graph. It is clear that if H is of size m, then oe(H) m. Recently Hartsfield and Smyth showed that for wheels W n of order n+1 and size m = 2n, oe(W n ) 2 Theta(m); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper we refine these results to show that for even n 4, oe(W n ) = n=2 + 2, while for odd n 5 we disprove a conjecture of Hartsfield and Smyth by showing that oe(W n ) = n. Labellings are given that achieve these minima

Patient-Specific Unicompartmental Knee Resurfacing Arthroplasty: Use of a Novel Interference Lock to Reduce Tibial Insert Micromotion and Backside Wear

Micromotion has long been associated with wear of polyethylene tibial inserts, potentially causing failure of unicompartmental knee replacement systems. One cause of micromotion is the locking mechanism between the undersurface of the polyethylene and the tibial tray. The objective of this study was to investigate the use of new novel lock designs for reducing the micromotion associated with a patient specific tibial implant. Micromotion occurring between the tibial insert and tibial tray was measured using the DVRT method for two new lock designs and compared to the micromotion measured for the prior generation lock design. In total, 18 samples were tested, six in each of the three designs: prior generation CLEARANCE lock design, new LINE-TO-LINE design, new INTERFERENCE design

Survey of certain valuations of graphs

The study of valuations of graphs is a relatively young part of graph theory. In this article we survey what is known about certain graph valuations, that is, labeling methods: antimagic labelings, edge-magic total labelings and vertex-magic total labelings

Vertex-magic total labelings of graphs

A vertex-magic total labeling of a graph with v vertices and e edges is de ned as a one-to-one map taking the vertices and edges onto the integers 1 � 2 � �v+ e with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex. Properties of these labelings are studied. It is shown how to construct labelings for several families of graphs, including cycles, paths, complete graphs of odd order and the complete bipartite graph Kn�n. It is also shown that labelings are impossible for some other classes of graphs

Increasing Fat Graft Retention in Irradiated Tissue after Preconditioning with External Volume Expansion

BACKGROUND: Fat grafting is an adjuvant that may improve the quality of radiation-damaged tissue. However, fat grafting for volume restoration in irradiated sites may be less effective because of a poorly vascularized fibrotic recipient bed. External volume expansion has emerged as a potential technique to prepare the recipient sites for improved survival of grafted fat. The authors previously demonstrated increased vasculature with external volume expansion stimulation of irradiated tissues. The authors now hypothesize that external volume expansion\u27s improvements in recipient-site vascularity will increase the volume retention and quality of fat grafts in fibrotic irradiated sites. METHODS: Athymic mice were irradiated until development of chronic radiation injury. Then, the irradiated site was stimulated by external volume expansion (external volume expansion group), followed by subcutaneous fat grafting. Grafts in an irradiated site without external volume expansion stimulation (irradiated control group) and grafts in a healthy nonirradiated (nonirradiated control group) site were used as controls. All grafts were monitored for 8 weeks and evaluated both histologically and by micro-computed tomography for analysis of volume retention. RESULTS: Hyperspectral imaging confirmed a 25 percent decrease in vascularity of irradiated tissue (irradiated control group) compared with nonirradiated tissue (nonirradiated control group). Grafts in the irradiated control group retained 11 percent less volume than grafts in the nonirradiated control group. The experimental external volume expansion group achieved a 20 percent (p = 0.01) increase in retained graft volume compared with the irradiated control group. CONCLUSIONS: External volume expansion stimulation can mitigate the effects of irradiation at the recipient site and in turn help preserve fat graft volume retention. Possible mechanisms include increased vascularity, adipogenic conversion, and increased compliance of a fibrotic recipient site