Abstract
AbstractThe determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 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, . . . or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.
Highlights
A graph is integral if the eigenvalues of its adjacency matrix are integers
The determinant Hosoya triangle mod gives rise to three in nite families of graphs, that are formed by complete product of two complete graphs with an empty graph
In this paper we study an in nite family of integral cographs associated with a combinatorial triangle
Summary
A graph is integral if the eigenvalues of its adjacency matrix are integers. These graphs are rare and the techniques used to nd them are quite complicated. In this paper we study an in nite family of integral cographs associated with a combinatorial triangle. These families have at most ve distinct eigenvalues. A graph (associated to the determinant Hosoya triangle) is integral if and only if its adjacency matrix is of size n= t+ From this triangle we obtain a rank two matrix S , depicted on the left side of (1). The main results of this paper show that the graphs that are generated from the matrices in that determinant Hosoya triangle mod are cographs of the form (Kn Km)∇Kr. We use G G to denote the disjoint union of G and G. We present a discussion on graphs associated to the Hosoya triangle, a triangular array where the entries are products of Fibonacci numbers
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