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

TL;DR

This paper explores the relationship between a specific type of graph's adjacency matrix and Fibonacci numbers, revealing how matrix powers relate to counting walks in the graph.

Contribution

It establishes a connection between the adjacency matrix of a certain graph and Fibonacci numbers, providing new insights into graph walks and matrix powers.

Findings

The (i,j)th entry of A^r counts walks of length r.

Matrix powers relate to Fibonacci sequence properties.

Provides a novel link between graph theory and Fibonacci numbers.

Abstract

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.