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Mathematics is used throughout the world as an essential tool in many fields. One of the new concepts in Graph Theory as the branch of mathematics is defined in this paper which is called the dominant local metric dimension. Let G be a connected graph. The ordered subsetWl = {w1, w2, w3, …, wn} ⊆ V(G) is called a dominant local resolving set of G if Wl is a local resolving set as well as a dominating set of G. A dominant local resolving set of G with minimum cardinality is called the dominant local basis of G. The cardinality of the dominant local basis of G is called the dominant local metric dimension of G and denoted by Ddiml(G). In this article, we present the methods for determining the dominant local metric dimension of graphs, characterize the dominant local metric dimension of graph G, and also determine the dominant local metric dimension of some particular classes of graphs.

This article covers the current understanding of human knee biomechanics and how these data are used in the assessment and development of knee replacement implants. Extensive and well corroborated data is available to the surgeon and engineer implant designer to aid in the design of new implant systems. The forces acting on the human knee joint have been measured experimentally from both external methods and permanently implanted telemetric devices. Motions of the knee are well understood for both the healthy knee as well as all commercially available total knee replacements today. Implant designers are now compelled to use these data in the design and testing of new implant systems. We now have 45 years in the modern age of knee replacement, with many lessons learned to-date, and repeating mistakes can now be avoided with just a fundamental understanding of the available published data on biomechanics and motion of the knee joint. To-date most implant designers have settled upon an averaging approach to the design of implant systems. We have now entered into an age where it is possible to customize implants for each individual patient yet respect all of the hard learned lessons of the past 45 years.

In this paper, the authors analysed metric dimensions of arbitrary graphs G★˜∧i=1|V(G)|Hi in which graphs G,H1,H2,…,H|V(G)| are non-trivial, G is connected, and ★˜ denotes generalized neighborhood corona operation. We found lower bounds of dim(G★˜∧i=1|V(G)|Hi) as function of dimA(Hi) where dimA(Hi) denotes adjacency metric dimensions of Hi. We also found upper bounds of dim(G★˜∧i=1|V(G)|Hi) when G does not contain pair of false twin vertices. Furthermore, we found a characteristic of dim(G★˜∧i=1|V(G)|Hi) which indicates that our lower bounds are strict., Neighborhood corona graph, metric dimension, resolving set.

The G be a connected graph with vertex set V(G) and edge set E(G). A subset S⊆V(G) is called a dominating set of G if for every vertex x in V(G)∖S, there exists at least one vertex u in S such that x is adjacent to u. An ordered set W⊆V(G) is called a resolving set of G, if every pair of vertices u and v in V(G) have distinct representation with respect to W. An ordered set S⊆V(G) is called a dominant resolving set of G, if S is a resolving set and also a dominating set of G. The minimum cardinality of dominant resolving set is called a dominant metric dimension of G, denoted by Ddim(G). In this paper, we investigate the dominant metric dimension of some particular class of graphs, the characterisation of graph with certain dominant metric dimension, and the dominant metric dimension of joint and comb products of graphs., Mathematics; Metric dimension; Resolving set; Dominating set; Dominant resolving set; Dominant metric dimension

Osteolysis ranks as the most significant cause of revision surgery in both total hip arthroplasty and total knee arthroplasty (TKA). The factors leading to osteolysis in TKA are unique and sometimes preventable. Changes in polyethylene manufacturing and implant design are striving to improve overall wear. In this review, we discuss osteolysis as it relates to TKAs. The etiology, diagnosis, contributing factors, and management are presented. The final section focuses on future improvements in TKA design, which may ultimately decrease the rate of osteolysis.

Let G=(V,E) be a finite graph, where |V|=n>=2 and |E|=e>=1. A vertex-magic total labeling is a bijection @l from V@?E to the set of consecutive integers {1,2,...,n+e} with the property that for every v@?V, @l(v)+@?"w"@?"N"("v")@l(vw)=h for some constant h. Such a labeling is strong if @l(V)={1,2,...,n}. In this paper, we prove first that the minimum degree of a strongly vertex-magic graph is at least two. Next, we show that if 2e>=10n^2-6n+1, then the minimum degree of a strongly vertex-magic graph is at least three. Further, we obtain upper and lower bounds of any vertex degree in terms of n and e. As a consequence we show that a strongly vertex-magic graph is maximally edge-connected and hamiltonian if the number of edges is large enough. Finally, we prove that semi-regular bipartite graphs are not strongly vertex-magic graphs, and we provide strongly vertex-magic total labeling of certain families of circulant graphs.

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L(w )= L(u )+ L(v). The sum number � (G) of a graph G =( V; E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is clear that � (G)6|E|. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that, over all graphs G =( V; E) with 5xed |V |?3 and |E|, the average of � (G) is at least |E |− 3|V |(log|V |)=[log(( |V | 2 )=|E|)] −| V |− 1. In other words, for most graphs, � (G) ∈ (|E|). c � 2001 Elsevier Science B.V. All rights reserved.

An almost Moore digraph is a digraph of diameter k⩾2 , maximum out-degree d⩾2 and order n=d+d 2 +⋯+d k , that is, one less than the Moore bound. It is easy to show that the out-degree of an almost Moore digraph is constant (=d) . In this note we prove that also the in-degree of an almost Moore digraph is constant (=d) , that is, every almost Moore digraph is diregular of degree d .

In this paper we prove that no wheel with the exception of W 4 can be a mod sum graph. We also give the unique (up to multiplication by positive integers) mod sum labelling of W 4 . We also prove that the symmetric complete bipartite graph K n . n is not a mod sum graph.