• Journal for History of Mathematics
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• v.21 no.3
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• pp.127-142
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• 2008

In this paper, we introduce four different approaches of proving Cayley formula, which counts the number of trees(acyclic connected simple graphs). The first proof was done by Cayley using recursive formulas. On the other hands the core idea of the other three proofs is the bijective method-find an one to one correspondence between the set of trees and a suitable family of combinatorial objects. Each of the three bijection gives its own generalization of Cayley formula. In particular, the last proof, done by Seo and Shin, has an application to computer science(theoretical computation), which is a typical example that pure mathematics supply powerful tools to other research fields.

• Journal for History of Mathematics
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• v.19 no.2
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• pp.125-140
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• 2006

In 1933, I. Schur introduced a Schur ring in connection with permutation group and regular subgroup. After then, it was studied mostly for purely group theoretical purposes. In 1970s, Klin and Poschel initiated its usage in the investigation of graphs, especially for Cayley and circulant graphs. Nowadays it is known that Schur ring is one of the best way to enumerate Cayley graphs. In this paper we study the origin of Schur ring back to 1933 and keep trace its evolution to graph theory and combinatorics.