mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Perron number is an

algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...

α which is real and exceeds 1, but such that its

conjugate element In mathematics, in particular field theory (mathematics), field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial (field theory), minimal polynomi ...

s are all less than α in

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

. For example, the larger of the two roots of the irreducible polynomial

x^ -3x + 1

is a Perron number. Perron numbers are named after

Oskar Perron Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differential ...

; the

Perron–Frobenius theorem In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive component ...

asserts that, for a

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...

with positive algebraic coefficients whose largest eigenvalue is greater than one, this eigenvalue is a Perron number. As a closely related case, the Perron number of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

is defined to be the

spectral radius In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...

of its

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...

. Any

Pisot number Charles Pisot (2 March 1910 – 7 March 1984) was a French mathematician. He is chiefly recognized as one of the primary investigators of the numerical set associated with his name, the Pisot–Vijayaraghavan numbers. He followed the classical p ...

Salem number In mathematics, a Salem number is a real algebraic integer ''α'' > 1 whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Dio ...

is a Perron number, as is the

Mahler measure In mathematics, the Mahler measure M(p) of a polynomial p(z) with complex coefficients is defined as M(p) = , a, \prod_ , \alpha_i, = , a, \prod_^n \max\, where p(z) factorizes over the complex numbers \mathbb as p(z) = a(z-\alpha_1)(z-\alph ...

of a monic integer polynomial.

References

* Algebraic numbers Graph invariants {{Numtheory-stub