Cross-ratio degrees count configurations of points z 1 , …</...

The Wiener index of a connected graph is the sum of all pairwise distances of vertices of the graph. In this paper, we consider the Wiener indices of trees with perfect matchings, characterizing the eight trees with smallest Wiener indices among all trees of order 2 ( 11) m m with perfect matchings.

Let B(2k) be the set of all bicyclic graphs on 2k(k¿ 2) vertices with perfect matchings. In this paper, we discuss some properties of the connected graphs with perfect matchings, and then determine graphs with maximal index in B(2k). c © 2004 Elsevier B.V. All rights reserved. MSC: 05c50

We determine the number of perfect matchings and forests in a family Tr,3 of triangulated prism graphs. These results show that the extremal number of perfect matchings in a planar graph with n vertices is bounded below by Ω ( 6 √ 7 + √ 37 n) = Ω (1.535) and the extremal number of forests contained in a planar graph with n vertices is bounded below by

In early 70s Berge conjectured that any bridgeless cubic graph contains five perfect matchings such that each edge belongs to at least one of them. In 1972 Fulkerson conjectured that, in fact, we can find six perfect matchings containing each edge exactly twice. By introducing the concept of an r-graph (a remarkable generalization of one of bridgeless cubic graph) Seymour in 1979 conjectured th...