Matrices in Hosoya Triangle

A triangular array where the entries are products of two Fibonacci numbers is Hosoya. The matrices within this triangle are of rank one (product of two vectors; located on the sides of the triangle). In this talk, we discuss properties and the behaviors of the eigenvalues, eigenvectors, characteristic polynomial, determinants, and their connection with graph theory. The non-zero eigenvalue is a combination of Lucas and Fibonacci numbers. In addition, these matrices are diagonalizable where the entries of the eigenvectors are points within the Hosoya Triangle. The components of the graphs (when matrices are seen mod 2) are complete graphs with loops and isolated vertices.