A characterization of consistent marked graphs

A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterzing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis

The number of labeled k-dimensional trees

AbstractThe graphs known as trees have natural analogues in higher dimensional simplicial complexes. As an extension of Cayley's formula nn−2, the number of these k-dimensional trees on n-labeled vertices is shown to be (nk) (kn−k2+1)n−k−2