Some inequalities on the norms of special matrices with generalized Tribonacci and generalized Pell Padovan sequences

In this paper some properties of generalized tribonacci and generalized Padovan sequence are presented. Also the Euclidean norms of circulant, $r$-circulant, semi-circulant and Hankle matrices with above mentioned sequences are calculated. The upper and lower bounds of spectral norms are also obtained.

Solution of the Diophantine Equation $ x_{1}x_{2}x_{3}cdots x_{m-1}=z^n $

This work determine the entire family of positive integer solutions of the diophantine equation. The solution is described in terms of $frac{(m-1)(m+n-2)}{2} $ or $frac{(m-1)(m+n-1)}{2}$ positive parameters depending on $n$ even or odd. We find the solution of a diophantine system of equations by using the solution of the diophantine equation. We generalized all the results of the paper [5].

Further Inequalities Between Vertex-Degree-Based Topological Indices

Continuing the recent work of L. Zhong and K. Xu [MATCH Commun. Math. Comput. Chem.71(2014) 627-642], we determine inequalities among several vertex-degree-based topological indices; first geometric-arithmetic index(GA), augmented Zagreb index (AZI), Randi$acute{c}$ index (R), atom-bond connectivity index (ABC), sum-connectivity index (X)and harmonic index (H).

On the Structure of Involutions and Symmetric Spaces of Quasi Dihedral Group

Let $G=QD_{8k}~$ be the quasi-dihedral group of order $8n$ and $theta$ be an automorphism of $QD_{8k}$ of finite order. The fixed-point set $H$ of $theta$ is defined as $H_{theta}=G^{theta}={xin G mid theta(x)=x}$ and generalized symmetric space $Q$ of $theta$ given by $Q_{theta}={gin G mid g=xtheta(x)^{-1}~mbox{for some}~xin G}.$ The characteristics of the sets $H$ and $Q$ have been calculated. It is shown that for any $H$ and $Q,~~H.Q eq QD_{8k}.$ the $H$-orbits on $Q$ are obtained under different conditions. Moreover, the formula to find the order of $v$-th root of unity in $mathbb{Z}_{2k}$ for $QD_{8k}$ has been calculated. The criteria to find the number of equivalence classes denoted by $C_{4k}$ of the involution automorphism has also been constructed. Finally, the set of twisted involutions $R=R_{theta}={~xin G~mid~theta(x)=x^{-1}}$ has been explored.