2,623 research outputs found
Spanning trees in a cactus
AbstractWe prove a best possible lower bound for the number of isomorphism classes into which all rooted spanning trees of a rooted cactus partition. We announce a best possible lower bound for the number of isomorphism classes into which all spanning trees of a cactus partition
Grassmann-Berezin Calculus and Theorems of the Matrix-Tree Type
We prove two generalizations of the matrix-tree theorem. The first one, a
result essentially due to Moon for which we provide a new proof, extends the
``all minors'' matrix-tree theorem to the ``massive'' case where no condition
on row or column sums is imposed. The second generalization, which is new,
extends the recently discovered Pfaffian-tree theorem of Masbaum and Vaintrob
into a ``Hyperpfaffian-cactus'' theorem. Our methods are noninductive, explicit
and make critical use of Grassmann-Berezin calculus that was developed for the
needs of modern theoretical physics.Comment: 23 pages, 2 figures, 3 references adde
On neighbour sum-distinguishing -edge-weightings of bipartite graphs
Let be a set of integers. A graph G is said to have the S-property if
there exists an S-edge-weighting such that any two
adjacent vertices have different sums of incident edge-weights. In this paper
we characterise all bridgeless bipartite graphs and all trees without the
-property. In particular this problem belongs to P for these graphs
while it is NP-complete for all graphs.Comment: Journal versio
Joint-tree model and the maximum genus of graphs
The vertex v of a graph G is called a 1-critical-vertex for the maximum genus
of the graph, or for simplicity called 1-critical-vertex, if G-v is a connected
graph and {\deg}M(G - v) = {\deg}M(G) - 1. In this paper, through the
joint-tree model, we obtained some types of 1-critical-vertex, and get the
upper embeddability of the Spiral Snm
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