A weighted graph zeta function involved in the Szegedy walk

We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.

On the Minimum Polynomial and Applications

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

Restructured class of estimators for population mean using an auxiliary variable under simple random sampling scheme

With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

Construction of orthomorphic $\mathrm{MDS}$ matrices with primitive characteristic polynomial

Проверочные матрицы линейных кодов с максимальным расстоянием ($\mathrm{MDS}$-матрицы) - важный элемент современных криптографических примитивов, обеспечивающий наилучшее рассеивание входных битов. В ряде работ изучались способы построения и описания $\mathrm{MDS}$-матриц для использования в низкоресурсной криптографии. Однако мало внимания уделялось влиянию приводимости предлагаемых $\mathrm{MDS}$-матриц, которая может позволить злоумышленнику использовать наличие нетривиальных инвариантных подпространств у соответствующих преобразований. В данной статье предлагаются некоторые методы построения $\mathrm{MDS}$-матриц с примитивными характеристическими многочленами, имеющие повышенную стойкость по отношению к атакам, основанным на инвариантных подпространствах.

A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.

Generalization of Multiplication M-Sequences over Fp and Its Reciprocal Sequences

Mp-Sequences or M-Sequence over Fp not used so much in current time as binary M-Sequences and it is pending with the difficult to construct there coders and decoders of Mp-Sequences further these reasons there is expensive values to construct them but the progress in the technical methods will be lead to fast using these sequences in different life’s ways, and these sequences give more collection of information and distribution them on the input and output links of the communication channels, building new systems with more complexity, larger period, and security. In current article we will study the construction of the multiplication Mp-Sequence {zn}and its linear equivalent, this sequences are as multiple two sequences, the first sequence{Sn}is an arbitrary Mp-Sequence and the second sequence {ζn} reciprocal sequence of the first sequence {Sn}, length of the sequence {zn}, period, orthogonal and the relations between the coefficients and roots of the characteristic polynomial of f(x) and it’s reciprocal polynomial g(x) and compare these properties with corresponding properties in M-Sequences.

Pseudo-random sequences of non-maximum length on shift registers with reducible and primitive polynomials

Inhomogeneous pseudo-random sequences of non-maximal length formed by shift registers with linear feedbacks based on a characteristic polynomial of degree n of the form ϕ(x)=ϕ1(x)ϕ2(x), where ϕ1(x) = x m1 ⊕ 1, and ϕ2(x) of degree m 2 is primitive (m 1 = 2 k , k is a positive integer, n = m 1 + m 2) are considered. Three schemes that are equivalent in terms of periodic sequence structures were considered. Of the greatest interest are the shift registers connected in an arbitrary way using a modulo-two adder, the feedbacks in which correspond to the multipliers ϕ1(x) and ϕ2(x) the polynomials ϕ(x). In this case, there is a complex process of forming output sequences, which involves both direct and inverse M-sequences. The statement about the singularity of the generated sequences at m 1 = 4 is proved, which is confirmed by their decimation with an index equal to the period of the primitive polynomial.

Estimates for solutions of systems of linear equations with circulant matrices

In the present paper we consider the problem to estimate a solution of the system of equations with a circulant matrix in uniform norm. We give the estimate for circulant matrices with diagonal dominance. The estimate is sharp. Based on this result and an idea of decomposition of the matrix into a product of matrices associated with factorization of the characteristic polynomial, we propose an estimate for any circulant matrix.

What we can learn from Bohemian Matrices?

This Maple Workbook explores a new topic in linear algebra, which is called "Bohemian Matrices". The topic is accessible to people who have had even just one linear algebra course, or have arrived at the point in their course where they have touched "eigenvalues". We use only the concepts of characteristic polynomial and eigenvalue. Even so, we will see some open questions, things that no-one knows for sure; even better, this is quite an exciting new area and we haven't even finished asking the easy questions yet! So it is possible that the reader will have found something new by the time they have finished going through this workbook. Reading this workbook is not like reading a paper: you will want to execute the code, and change things, and try alternatives. You will want to read the code, as well. I have tried to make it self-explanatory. We will begin with some pictures, and then proceed to show how to make such pictures using Maple (or, indeed, many other computational tools). Then we start asking questions about the pictures, and about other things.