Computer Methodologies for the Comparison of Some Efficient Derivative FreeSimultaneous Iterative Methods for Finding Roots of Non-Linear Equations

In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine.Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU time-methodology.Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods.Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples.Numerical test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn, a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H5,n by partially answering to an open problem proposed in Ⅱ. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88： 43-57]. We prove that these classes of regular graphs have constant metric dimension.

Resolvability in Circulant Graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V（G） there is a vertex w E W such that d（u, w） ≠ d（v, w）. A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim（G）. For a vertex u of G and a subset S of V（G）, the distance between u and S is the number minses d（u,s）. A k-partition II = {$1,$2,..., Sk} of V（G） is called a resolving partition if for every two distinct vertices u, v E V（G） there is a set Si in H such that d（u, Si） ≠ d（v, Si）. The minimum k for which there is a resolving k-partition of V（G） is called the partition dimension of G, denoted by pd（G）. The circulant graph is a graph with vertex set Zn, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i - j （rood n） E C, where C C Zn has the property that C = -C and 0 ￠ C. The circulant graph is denoted by Xn,△ where A = ｜C｜. In this paper, we study the metric dimension of a family of circulant graphs Xn,3 with connection set C = {1, 3, n - 1} and prove that dim（Xn,3） is independent of choice of n by showing that dim（Xn,a） = {3 for all n ≡0 （mod 4）,4 for all n≡2 （mod4）.We also study the partition dimension of a family of circulant graphs Xm,4 with connection set C = {±1, ±2} and prove that pd（Xn,4） is independent of choice of n and show that pd（X5,4） = 5 and pd（Xn,a） ={ 3 for all odd n≥9,4 for all even n≥6 and n=7.

On the Constant Metric Dimension of Generalized Petersen Graphs P(n,4)

In this paper, we consider the family of generalized Petersen graphs P（n,4）. We prove that the metric dimension of P（n, 4） is 3 when n = 0 （mod 4）, and is 4 when n = 4k ＋ 3 （k is even）.For n = 1,2 （mod 4） and n = 4k ＋ 3 （k is odd）, we prove that the metric dimension of P（n,4） is bounded above by 4. This shows that each graph of the family of generalized Petersen graphs P（n, 4） has constant metric dimension.

Braids with Trivial Simple Centralizer

In this paper, we prove that the n-simple braid divisible by the generators xi for all 2 ≤ i ≤n - 2 has trivial simple centralizer. Consequently, the commuting graph defined on the set of simple braids is disconnected. We also prove that the graph has one major component.