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sci.math FAQ: Erdos Number

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Archive-name: sci-math-faq/erdosnumber
Last-modified: February 20, 1998
Version: 7.5

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                                Erdos Number
   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
   indefatigable traveller.
   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 under the directory pub/math/erdos or on the Web
   at 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                                                      Assistant Professor	
Faculty of Computer Science                  University of New Brunswick

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