1,748 research outputs found
On the spanning tree packing number of a graph: a survey
AbstractThe spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes
Fully Dynamic Effective Resistances
In this paper we consider the \emph{fully-dynamic} All-Pairs Effective
Resistance problem, where the goal is to maintain effective resistances on a
graph among any pair of query vertices under an intermixed sequence of edge
insertions and deletions in . The effective resistance between a pair of
vertices is a physics-motivated quantity that encapsulates both the congestion
and the dilation of a flow. It is directly related to random walks, and it has
been instrumental in the recent works for designing fast algorithms for
combinatorial optimization problems, graph sparsification, and network science.
We give a data-structure that maintains -approximations to
all-pair effective resistances of a fully-dynamic unweighted, undirected
multi-graph with expected amortized
update and query time, against an oblivious adversary. Key to our result is the
maintenance of a dynamic \emph{Schur complement}~(also known as vertex
resistance sparsifier) onto a set of terminal vertices of our choice.
This maintenance is obtained (1) by interpreting the Schur complement as a
sum of random walks and (2) by randomly picking the vertex subset into which
the sparsifier is constructed. We can then show that each update in the graph
affects a small number of such walks, which in turn leads to our sub-linear
update time. We believe that this local representation of vertex sparsifiers
may be of independent interest
On Jacobian group of the -graph
In the present paper we compute the Jacobian group of -graph
The notion of -graph continues the list of
families of -, - and -graphs well-known in the graph theory. In
particular, graph is isomorphic to discrete torus
It this case, the structure of the Jacobian group will be find
explicitly.Comment: arXiv admin note: text overlap with arXiv:2111.0430
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