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Subject: sci.math FAQ: Erdos Number
This article was archived around: 17 Feb 2000 22:52:00 GMT
Last-modified: February 20, 1998
Form an undirected graph where the vertices are academics, and an edge
connects academic X to academic Y if X has written a paper with Y. The
Erdos number of X is the length of the shortest path in this graph
connecting X with Erdos.
Erdos has Erdos number 0. Co-authors of Erdos have Erdos number 1.
Einstein has Erdos number 2, since he wrote a paper with Ernst Straus,
and Straus wrote many papers with Erdos.
The Extended Erdos Number applies to co-authors of Erdos. For People
who have authored more than one paper with Erdos, their Erdos number
is defined to be 1/# papers-co-authored.
Why people care about it?
Nobody seems to have a reasonable answer...
Who is Paul Erdos?
Paul Erdos was an Hungarian mathematician. He obtained his PhD from
the University of Manchester and spent most of his efforts tackling
"small" problems and conjectures related to graph theory,
combinatorics, geometry and number theory.
He was one of the most prolific publishers of papers; and was also and
Paul Erdvs died on September 20, 1996.
At this time the number of people with Erdos number 2 or less is
estimated to be over 4750, according to Professor Jerrold W. Grossman
archives. These archives can be consulted via anonymous ftp at
vela.acs.oakland.edu under the directory pub/math/erdos or on the Web
at http://www.acs.oakland.edu/ grossman/erdoshp.html. At this time it
contains a list of all co-authors of Erdos and their co-authors.
On this topic, he writes
Let E_1 be the subgraph of the collaboration graph induced by
people with Erdos number 1. We found that E_1 has 451 vertices and
1145 edges. Furthermore, these collaborators tended to collaborate
a lot, especially among themselves. They have an average of 19
other collaborators (standard deviation 21), and only seven of them
collaborated with no one except Erdos. Four of them have over 100
co-authors. If we restrict our attention just to E_1, we still find
a lot of joint work. Only 41 of these 451 people have collaborated
with no other persons with Erdos number 1 (i.e., there are 41
isolated vertices in E_1), and E_1 has four components with two
vertices each. The remaining 402 vertices in E_1 induce a connected
subgraph. The average vertex degree in E_1 is 5, with a standard
deviation of 6; and there are four vertices with degrees of 30 or
higher. The largest clique in E_1 has seven vertices, but it should
be noted that six of these people and Erdos have a joint
seven-author paper. In addition, there are seven maximal 6-cliques,
and 61 maximal 5-cliques. In all, 29 vertices in E_1 are involved
in cliques of order 5 or larger. Finally, we computed that the
diameter of E_1 is 11 and its radius is 6.
Three quarters of the people with Erdos number 2 have only one
co-author with Erdos number 1 (i.e., each such person has a unique
path of length 2 to p). However, their mean number of Erdos number
1 co-authors is 1.5, with a standard deviation of 1.1, and the
count ranges as high as 13.
Folklore has it that most active researchers have a finite, and
fairly small, Erdos number. For supporting evidence, we verified
that all the Fields and Nevanlinna prize winners during the past
three cycles (1986--1994) are indeed in the Erdos component, with
Erdos number at most 9. Since this group includes people working in
theoretical physics, one can conjecture that most physicists are
also in the Erdos component, as are, therefore, most scientists in
general. The large number of applications of graph theory to the
social sciences might also lead one to suspect that many
researchers in other academic areas are included as well. We close
with two open questions about C, restricted to mathematicians, that
such musings suggest, with no hope that either will ever be
answered satisfactorily: What is the diameter of the Erdos
component, and what is the order of the second largest component?
Caspar Goffman. And what is your Erdos number? American Mathematical
Monthly, v. 76 (1969), p. 791.
Tom Odda (alias for Ronald Graham) On Properties of a Well- Known
Graph, or, What is Your Ramsey Number? Topics in Graph Theory (New
York, 1977), pp. [166-172].
Alex Lopez-Ortiz firstname.lastname@example.org
http://www.cs.unb.ca/~alopez-o Assistant Professor
Faculty of Computer Science University of New Brunswick