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Graph Powers The investigation of the asymptotic behaviour of various parameters of powers of a fixed
 

Summary: Graph Powers
Noga Alon
Abstract
The investigation of the asymptotic behaviour of various parameters of powers of a fixed
graph leads to many fascinating problems, some of which are motivated by questions in infor-
mation theory, communication complexity, geometry and Ramsey theory. In this survey we
discuss these problems and describe the techniques used in their study which combine combi-
natorial, geometric, probabilistic and linear-algebra tools.
1 Graph Powers
There are several known distinct ways to define the powers of a fixed graph. The n-th AND power
of an undirected graph G = (V, E) is the graph denoted by Gn whose vertex set is V n in which
distinct vertices (x1 . . . xn) and (x1 . . . xn) are connected if {xi, xi} E for all i {1, 2, . . . , n} such
that xi = xi. The n-th OR power of G is the graph denoted Gn whose vertex set is V n in which
distinct vertices (x1 . . . xn) and (x1 . . . xn) are connected if distinct xi and xi are connected in G
for some i {1, 2, . . . , n}.
The study of the asymptotic behaviour of various parameters of these powers of a fixed graph
G, as well as their behaviour for similarly defined powers of directed and undirected graphs, is
motivated by questions in various areas and leads to many intriguing problems. These are discussed
in the following sections, in which we focus our attention mainly to the open problems in the area,
and only briefly describe the known results and proof techniques. Proven and disproven conjectures

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics