405 research outputs found
The terms in Lucas sequences divisible by their indices
For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as
usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are
either integers or conjugate quadratic integers, we describe the set of indices
n for which n divides u_n and also the set of indices n for which n divides
v_n. Building on earlier work, particularly that of Somer, we show that the
numbers in these sets can be written as a product of a so-called basic number,
which can only be 1, 6 or 12, and particular primes, which are described
explicitly. Some properties of the set of all primes that arise in this way is
also given, for each kind of sequence
Integer symmetric matrices having all their eigenvalues in the interval [-2,2]
We completely describe all integer symmetric matrices that have all their
eigenvalues in the interval [-2,2]. Along the way we classify all signed
graphs, and then all charged signed graphs, having all their eigenvalues in
this same interval. We then classify subsets of the above for which the integer
symmetric matrices, signed graphs and charged signed graphs have all their
eigenvalues in the open interval (-2,2).Comment: 33 pages, 18 figure
Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]
The adjacency matrices of graphs form a special subset of the set of all
integer symmetric matrices. The description of which graphs have all their
eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most
2) has been known for several decades. In 2007 we extended this classification
to arbitrary integer symmetric matrices.
In this paper we turn our attention to symmetrizable matrices. We classify
the connected nonsymmetric but symmetrizable matrices which have entries in
that are maximal with respect to having all their eigenvalues in [-2,2].
This includes a spectral characterisation of the affine and finite Dynkin
diagrams that are not simply laced (much as the graph result gives a spectral
characterisation of the simply laced ones).Comment: 20 pages, 11 figure
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