Journal of Applied Mathematics
Volume 2012 (2012), Article ID 423163, 14 pages
Research Article

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

Department of Mathematics, Science Faculty, Selcuk University, 42250 Konya, Turkey

Received 9 February 2012; Accepted 26 March 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Fatih Yılmaz and Durmuş Bozkurt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 𝐴 𝑚 ( 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 𝑚 th 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.