Families of Integral Cographs within a Triangular Arrays

TL;DR

This paper explores families of integral cographs derived from the determinant Hosoya triangle modulo 2, characterizing their spectral properties and extending results to the entire triangle.

Contribution

It provides a necessary and sufficient condition for graphs in these families to be integral and analyzes their eigenvalues and regularity properties.

Findings

Graphs have at most five distinct eigenvalues.

All graphs are either d-regular with d=2,4,6,... or almost-regular.

Some graphs are Laplacian integral.

Abstract

The \emph{determinant Hosoya triangle}, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle $\bmod \,2$ gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral. Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either $d$-regular graphs with $d=2,4,6,\dots $ or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.