88 research outputs found
Maximally connected and super arc-connected Bi-Cayley digraphs
Let X=(V, E) be a digraph. X is maximally connected, if \kappa(X)=\delta(X).
X is maximally arc-connected, if \lambda(X)=\delta(X). And X is super
arc-connected, if every minimum arc-cut of X is either the set of inarcs of
some vertex or the set of outarcs of some vertex. In this paper, we will prove
that the strongly connected Bi-Cayley digraphs are maximally connected and
maximally arc-connected, and the most of strongly connected Bi-Cayley digraphs
are super arc-connected.Comment: 11pages,0 figure
Arc-connectivity and super arc-connectivity of mixed Cayley digraph
A digraph X=(V, E) is max-\lambda, if \lambda(X)=\delta(X). A digraph X is
super-\lambda if every minimum cut of X is either the set of inarcs of some
vertex or the set of outarcs of some vertex. In this paper, we will prove that
for all but a few exceptions, the strongly connected mixed Cayley digraphs are
max-\lambda and super-\lambda.Comment: 25pages,9 figure
Complex Balanced Spaces
In this paper, the concept of balanced manifolds is generalized to reduced
complex spaces: the class B and balanced spaces. Compared with the case of
Kahlerian, the class B is similar to the Fujiki class C and the balanced space
is similar to the Kahler space. Some properties about these complex spaces are
obtained, and the relations between the balanced spaces and the class B are
studied.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1610.0715
On restricted edge-connectivity of half-transitive multigraphs
Let be a multigraph (it has multiple edges, but no loops). The edge
connectivity, denoted by , is the cardinality of a minimum edge-cut
of . We call maximally edge-connected if , and
super edge-connected if every minimum edge-cut is a set of edges incident with
some vertex. The restricted edge-connectivity of is the
minimum number of edges whose removal disconnects into non-trivial
components. If achieves the upper bound of restricted
edge-connectivity, then is said to be -optimal. A bipartite
multigraph is said to be half-transitive if its automorphism group is
transitive on the sets of its bipartition. In this paper, we will characterize
maximally edge-connected half-transitive multigraphs, super edge-connected
half-transitive multigraphs, and -optimal half-transitive
multigraphs
The -spectral radius of graphs with given degree sequence
Let be a graph with adjacency matrix , and let be the
diagonal matrix of the degrees of . For any real , write
for the matrix This
paper presents some extremal results about the spectral radius
of that generalize previous results about
and . In this paper, we give some
results on graph perturbation for -matrix with . As
applications, we characterize all extremal trees with the maximum
-spectral radius in the set of all trees with prescribed degree
sequence firstly. Furthermore, we characterize the unicyclic graphs that have
the largest -spectral radius for a given unicycilc degree sequence
On the sizes of -edge-maximal -uniform hypergraphs
Let be a hypergraph, where is a set of vertices and is a
set of non-empty subsets of called edges. If all edges of have the same
cardinality , then is a -uniform hypergraph; if consists of all
-subsets of , then is a complete -uniform hypergraph, denoted by
, where . A -uniform hypergraph is
-edge-maximal if every subhypergraph of with
has edge-connectivity at most , but for any edge , contains at least one subhypergraph with
and edge-connectivity at least . In this paper, we obtain the lower bounds
and the upper bounds of the sizes of -edge-maximal hypergraphs.
Furthermore, we show that these bounds are best possible. Thus prior results in
[Y.Z. Tian, L.Q. Xu, H.-J. Lai, J.X. Meng, On the sizes of -edge-maximal
-uniform hypergraphs, arXiv:1802.08843v3] are extended.Comment: arXiv admin note: text overlap with arXiv:1802.0884
On the sizes of -edge-maximal -uniform hypergraphs
Let be a hypergraph, where is a set of vertices and is a
set of non-empty subsets of called edges. If all edges of have the same
cardinality , then is a -uniform hypergraph; if consists of all
-subsets of , then is a complete -uniform hypergraph, denoted by
, where . A hypergraph is called a subhypergraph of
if and . A -uniform hypergraph
is -edge-maximal if every subhypergraph of has
edge-connectivity at most , but for any edge ,
contains at least one subhypergraph with edge-connectivity at least
.
Let and be integers with and , and let be
the largest integer such that . That is, is the
integer satisfies . We prove that if is
a -uniform -edge-maximal hypergraph such that , then
() , and this bound is best possible;
() , and
this bound is best possible.
This extends former results in [8] and [6]
Lower Bound for the Simplicial Volume of Closed Manifolds Covered by
We estimate the upper bound for the -norm of the volume form
on seen as a class in
.
This gives the lower bound for the simplicial volume of closed Riemennian
manifolds covered by .
The proof of these facts yields an algorithm to compute the lower bound of
closed Riemannian manifolds covered by
Connectivity keeping stars or double-stars in 2-connected graphs
In [W. Mader, Connectivity keeping paths in -connected graphs, J. Graph
Theory 65 (2010) 61-69.], Mader conjectured that for every positive integer
and every finite tree with order , every -connected, finite graph
with contains a subtree
isomorphic to such that is -connected. In the same paper,
Mader proved that the conjecture is true when is a path. Diwan and Tholiya
[A.A. Diwan, N.P. Tholiya, Non-separating trees in connected graphs, Discrete
Math. 309 (2009) 5235-5237.] verified the conjecture when . In this paper,
we will prove that Mader's conjecture is true when is a star or double-star
and
Nonseparating trees in 2-connected graphs and oriented trees in strongly connected digraphs
Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive
integer and every finite tree with order , every -connected,
finite graph with contains a
subtree isomorphic to such that is -connected. The
conjecture has been verified for paths, trees when , and stars or
double-stars when . In this paper we verify the conjecture for two classes
of trees when .
For digraphs, Mader [J. Graph Theory 69 (2012) 324-329] conjectured that
every -connected digraph with minimum semi-degree
for a positive integer
has a dipath of order with . The conjecture
has only been verified for the dipath with , and the dipath with and
. In this paper, we prove that every strongly connected digraph with
minimum semi-degree contains
an oriented tree isomorphic to some given oriented stars or double-stars
with order such that is still strongly connected
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