Undergraduate Mathematics  
Research Group 

Research Announcement of the WSU graduate and undergraduate research group, 1995.

Definition: The graph

is called the flower of the type k, m and will be denoted Jk,m. Recall, that a tree G on h vertices is called perfectible if there exist integer weights wi > 2 on its vertices such that the matrix

is postitive definite with determinant equal to 1. A perfectible graph is minimal if it does not contain any perfictible subgraphs.
Theorem: The following is the complete list of perfectible flowers Jk,m with m > 2 (pictured with one set of perfect weights). A flower of Jk,m m > 2 is perfectible U G contains one of these five graphs.
 
 
 
 

Connection to number theory: Let

be a perfectly weghted flower. Then (and only then) integers n1, ..., nk, wo, ..., wm satisfy the Diophantine unit fraction equation:

where P/Q is the continued fraction

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We now know all perfectible trees except

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Last changed: November 3, 1997
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