 
Summary: The Moore bound for irregular graphs
Noga Alon
Shlomo Hoory
Nathan Linial
February 22, 2002
Abstract
What is the largest number of edges in a graph of order n and girth g? For d
regular graphs, essentially the best known answer is provided by the Moore bound.
This result is extended here to cover irregular graphs as well, yielding an affirmative
answer to an old open problem ([4] p.163, problem 10).
What is the maximal number of edges in a graph with n vertices and girth g? Put
differently, what is the least number of vertices n = n(d, g) in a graph of girth g and an
average degree d?
If d is an integer, it is natural to compare the lower bound on n, to what is known for
dregular graphs. The lower bound for dregular graphs arises from the fact the ball of
radius g1
2
around a vertex or an edge (depending on the parity of g) is a tree. This
simple argument yields the socalled Moore bound denoted by n0(d, g) (see [2], p. 180),
where:
