On a Selberg–Schur Integral

On a Zonal Polynomial Integral

A survey of results on integral trees and integral graphs

A graph G is called integral if all zeros of the characteristic polynomial P (G, x) are integers. Our purpose is to determine or to characterize which graphs are integral. This problem was posed by Harary and Schwenk in 1974. In general, the problem of characterizing integral graphs seems to be very difficult. Thus, it makes sense to restrict our investigations to some interesting families of g...

A lecture on integral geometry

Integral geometry originated from geometric probability. It studies random geometric objects in a probability space endowed with a measure that is invariant under a group of transformations. The geometric objects are points, lines, planes, solids, curves, surfaces, geodesics, etc. Works of Crofton, Poincaré, Sylvester, and others set up the early stage of integral geometry. Poincaré defined the...

A Survey on Integral Graphs

Throughout this paper a graph G is assumed to be simple, i.e. a finite undirected graph without loops or multiple edges. Therefore, the characteristic polynomial of (the adjacency matrix of) G, denoted by PG(λ), has only real zeroes and this family of eigenvalues (the spectrum of G) will be represented as (λ1, λ2, . . . , λn), where λ1 ≥ λ2 ≥ · · · ≥ λn or in the form μ1 1 , μ2 2 , . . . , μm m...

A Course on Integral Domains

My son who is in the 4 grade is learning about prime numbers and cancelling prime numbers in order to reduce fractions into lowest forms. I have told him that every number (positive integer) can be expressed as a product of primes, and surely along the road, his teachers will confirm this. We will consider this property in integral domains. We say that a divides b in the domain R, and write a |...