The adjacency matrix of one type of directed graph and the Jacobsthal numbers and their determinantal representation
type: Item
instance of: scholarly article
title: The adjacency matrix of one type of directed graph and the Jacobsthal numbers and their determinantal representation
zbMATH Open document ID: 1258.05079
DOI: 10.1155/2012/423163
author: Durmuş Bozkurt
author: Fatih Yılmaz
published in: Journal of Applied Mathematics
publication date: 2012-11-15T00:00:00Z
review text: Summary: 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.
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 6106209
zbMATH Keywords: adjacency matrix
zbMATH Keywords: directed graph
zbMATH Keywords: number of walks
zbMATH Keywords: Jacobsthal number
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/423163
OpenAlex ID: W2125591649
Wikidata QID: Q58905867
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DBLP publication ID: journals/jam/YilmazB12
description: scientific article; zbMATH DE number 6106209