Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the (𝑖,𝑗) entry of 𝐴𝑚 (𝐴 is adjacency matrix) is equal to the number of walks of length 𝑚 from vertex 𝑖 to vertex 𝑗, we show that elements of 𝑚th positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers.

/pdf/the-adjacency-matrix-of-one-type-of-directed-graph-and-the-57be697095.pdf

The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation

In this paper, we firstly derive a general expression for the entries of the 
 
 m
 
 th (
 
 m
 ∈
 ℕ
 
 ) power for two certain types of tridiagonal matrices of arbitrary order. Secondly, we present a method for computing the positive integer powers of the anti-tridiagonal matrix corresponding to these matrices. Also, we give Maple 18 procedures in order to verify our calculations.

Positive Integer Powers of Certain Tridiagonal Matrices and Corresponding Anti-Tridiagonal Matrices

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we investigate relationships between one type of graph and well-known Fibonacci sequence. In this content, we consider the adjacency matrix of one type of graph with 2k (k=1,2,...) vertices. It is also known that for any positive integer r, the (i,j)th entry of A^{r} (A is the adjacency matrix of the graph) is just the number of walks from vertex i to vertex j, that use exactly k edges.

The adjacency matrix of one type of graph and the Fibonacci numbers

In this paper, we obtain the eigenvalues and eigenvectors of a certain complex tridiagonal matrix family in terms of the Chebyshev polynomials of the first kind.

/pdf/eigenvalues-and-eigenvectors-of-a-certain-complex-d6hjhnyju6.pdf

Eigenvalues and eigenvectors of a certain complex tridiagonal matrix family

In this paper, we derive the general expression of the r th power (r ∈ N) for one type of tridiagonal matrix.

/pdf/a-formula-for-computing-integer-powers-for-one-type-of-57ty0te19r.pdf

A formula for computing integer powers for one type of tridiagonal matrix

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A formula for computing integer powers for one type of tridiagonal matrix