Summary: The Moore bound for irregular graphs
February 22, 2002
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 ( 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
d-regular graphs. The lower bound for d-regular graphs arises from the fact the ball of
around a vertex or an edge (depending on the parity of g) is a tree. This
simple argument yields the so-called Moore bound denoted by n0(d, g) (see , p. 180),