Journal of Linear and Topological Algebra
Volume:6 Issue: 1, Winter 2017

‎Let $G$ be a graph without an isolated vertex‎, ‎the normalized Laplacian matrix $tilde{mathcal{L}}(G)$‎ ‎is defined as $tilde{mathcal{L}}(G)=mathcal{D}^{-frac{1}{2}}mathcal{L}(G)mathcal{D}^{-frac{1}{2}}$‎, where ‎$mathcal{D}$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎G‎$‎‎. ‎The eigenvalues of‎ $tilde{mathcal{L}}(G)$ are ‎called as ‎the ‎normalized Laplacian eigenvalues of $G$‎. ‎In this paper‎, ‎we obtain the normalized Laplacian spectrum of two new types of join graphs‎. ‎In continuing‎, ‎we determine the integrality of normalized Laplacian eigenvalues of graphs‎. ‎Finally‎, ‎the normalized Laplacian energy and degree Kirchhoff index of these new graph ‎products‎ are derived‎.

In this paper, An effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (RBFs). We present an algorithm based on interpolation by radial basis functions including multiquadratics (MQs), using Legendre-Gauss-Lobatto nodes and weights. Also a theorem is proved for convergence of the algorithm. Some numerical examples are presented and results are compared to the analytical solution and Triangular functions (TF), Delta basis functions (DFs), block-pulse functions , sinc fucntions, Adomian decomposition, computational, Haar wavelet and direct methods to demonstrate the validity and applicability of the proposed method.

In this paper we prove a unique common coupled fixed point theorem for two pairs of $w$-compatible mappings in $S_b$-metric spaces satisfying a contrctive type condition. We furnish an example to support our main theorem. We also give a corollary for Junck type maps.

In this manuscript, we prove some coupled fixed point theorems for two pairs of self mappings satisfying contractive conditions of integral type in generalized metric spaces. We furnish suitable illustrative examples. In this manuscript, we prove some coupled fixed point theorems for two pairs of self mappings satisfying contractive conditions of integral type in generalized metric spaces. We furnish suitable illustrative examples.

The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Sigma^{n-2}_{k=0} Sigma^{n-2-k}_{i=0} x^kdelta(x)x^idelta(x)x^{n-2-k-i}$$ is fulfilled for all $xin mathcal{R}$. If $delta(1) = 0$, then $d$ is a Jordan $delta$-double derivation. In particular, if $mathcal{R}$ is a semiprime algebra and further, $delta^2(x^2) = delta^2(x)x + xdelta^2(x) + 2(delta(x))^2$ holds for all $xin mathcal{R}$, then $d-frac{1}{2}delta^2$ is an ordinary derivation on $mathcal{R}$.

Given four complex matrices $A$‎, ‎$B$‎, ‎$C$ and $D$ where $Ainmathbb{C}^{ntimes n}$‎ ‎and $Dinmathbb{C}^{mtimes m}$ and let the matrix $left(begin{array}{cc}‎ A & B ‎ C & D‎ end{array} right)$ be a normal matrix and‎ assume that $lambda$ is a given complex number‎ ‎that is not eigenvalue of matrix $A$‎. ‎We present a method to calculate the distance norm (with respect to 2-norm) from $D$‎ to the set of matrices $X in C^{m times m}$ such that‎, ‎$lambda$ be a multiple‎ eigenvalue of matrix $left(begin{array}{cc}‎ A & B ‎ C & X‎ end{array} right)$‎. ‎We‎ also find the nearest matrix $X$ to the matrix $D$‎.

Best approximation results provide an approximate solution to the fixed point equation $Tx=x$, when the non-self mapping $T$ has no fixed point. In particular, a well-known best approximation theorem, due to Fan cite{5}, asserts that if $K$ is a nonempty compact convex subset of a Hausdorff locally convex topological vector space $E$ and $T:Krightarrow E$ is a continuous mapping, then there exists an element $x$ satisfying the condition $d(x,Tx)=inf {d(y,Tx):yin K}$, where $d$ is a metric on $E$. Recently, Hussain et al. (Abstract and Applied Analysis, Vol. 2014, Article ID 837943) introduced proximal contractive mappings and established certain best proximity point results for these mappings in $G$-metric spaces. The aim of this paper is to introduce certain new classes of auxiliary functions and proximal contraction mappings and establish best proximity point theorems for such kind of mappings in $G$-metric spaces. As consequences of these results, we deduce certain new best proximity and fixed point results in $G$-metric spaces. Moreover, we present certain examples to illustrate the usability of the obtained results.