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

Fatih Yılmaz and Durmuş Bozkurt

Journal of Applied Mathematics, 2012, vol. 2012, 1-14

Abstract:

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 ( ð ‘– , ð ‘— ) entry of ð ´ ð ‘š ( ð ´ is adjacency matrix) is equal to the number of walks of length ð ‘š from vertex ð ‘– to vertex ð ‘— , we show that elements of ð ‘š 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.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:423163

DOI: 10.1155/2012/423163

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