50 research outputs found
Graphical condensation of plane graphs: a combinatorial approach
The method of graphical vertex-condensation for enumerating perfect matchings
of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003),
267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57)
and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this
paper, by a purely combinatorial method some explicit identities on graphical
vertex-condensation for enumerating perfect matchings of plane graphs (which do
not need to be bipartite) are obtained. As applications of our results, some
results on graphical edge-condensation for enumerating perfect matchings are
proved, and we count the sum of weights of perfect matchings of weighted Aztec
diamond.Comment: 13 pages, 5 figures. accepted by Theoretial Computer Scienc
Enumeration of subtrees of trees
Let be a weighted tree. The weight of a subtree of is defined
as the product of weights of vertices and edges of . We obtain a
linear-time algorithm to count the sum of weights of subtrees of . As
applications, we characterize the tree with the diameter at least , which
has the maximum number of subtrees, and we characterize the tree with the
maximum degree at least , which has the minimum number of subtrees.Comment: 20 pages, 11 figure
The determinant of q-distance matrices of trees and two quantities relating to permutations
In this paper we prove that two quantities relating to the length of
permutations defined on trees are independent of the structures of trees. We
also find that these results are closely related to the results obtained by
Graham and Pollak (Bell System Tech. J. 50(1971) 2495--2519) and by Bapat,
Kirkland, and Neumann (Linear Alg. Appl. 401(2005) 193--209).Comment: 12 pages, 1 figur