2,251 research outputs found
Bounds on the edge-Wiener index of cacti with vertices and cycles
The edge-Wiener index of a connected graph is the sum of
distances between all pairs of edges of . A connected graph is said to
be a cactus if each of its blocks is either a cycle or an edge. Let
denote the class of all cacti with vertices and
cycles. In this paper, the upper bound and lower bound on the edge-Wiener index
of graphs in are identified and the corresponding extremal
graphs are characterized
Antimagic orientations of disconnected even regular graphs
A of a digraph with arcs is a bijection from the set of
arcs of to . A labeling of is if no two
vertices in have the same vertex-sum, where the vertex-sum of a vertex for a labeling is the sum of labels of all arcs entering minus
the sum of labels of all arcs leaving . An antimagic orientation of a
graph is if has an antimagic labeling. Hefetz,
Mtze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the
question: Does every graph admits an antimagic orientation? It had been proved
that for any integer , every 2-regular graph with at most two odd
components has an antimagic orientation. In this paper, we consider the
2-regular graph with many odd components. We show that every 2-regular
graph with any odd components has an antimagic orientation provide each odd
component with enough order
The generalized connectivity of some regular graphs
The generalized -connectivity of a graph is a
parameter that can measure the reliability of a network to connect any
vertices in , which is proved to be NP-complete for a general graph . Let
and denote the maximum number of
edge-disjoint trees in such that
for any and . For an integer with , the {\em generalized
-connectivity} of a graph is defined as and .
In this paper, we study the generalized -connectivity of some general
-regular and -connected graphs constructed recursively and obtain
that , which attains the upper bound of
[Discrete Mathematics 310 (2010) 2147-2163] given by Li {\em et al.} for
. As applications of the main result, the generalized -connectivity
of many famous networks such as the alternating group graph , the
-ary -cube , the split-star network and the
bubble-sort-star graph etc. can be obtained directly.Comment: 19 pages, 6 figure
The -good neighbour diagnosability of hierarchical cubic networks
Let be a connected graph, a subset is called an
-vertex-cut of if is disconnected and any vertex in has
at least neighbours in . The -vertex-connectivity is the size
of the minimum -vertex-cut and denoted by . Many
large-scale multiprocessor or multi-computer systems take interconnection
networks as underlying topologies. Fault diagnosis is especially important to
identify fault tolerability of such systems. The -good-neighbor
diagnosability such that every fault-free node has at least fault-free
neighbors is a novel measure of diagnosability. In this paper, we show that the
-good-neighbor diagnosability of the hierarchical cubic networks
under the PMC model for and the model for is , respectively
Fault-tolerance of balanced hypercubes with faulty vertices and faulty edges
Let (resp. ) be the set of faulty vertices (resp. faulty edges)
in the -dimensional balanced hypercube . Fault-tolerant Hamiltonian
laceability in with at most faulty edges is obtained in [Inform.
Sci. 300 (2015) 20--27]. The existence of edge-Hamiltonian cycles in
for are gotten in [Appl. Math. Comput. 244 (2014) 447--456].
Up to now, almost all results about fault-tolerance in with only faulty
vertices or only faulty edges. In this paper, we consider fault-tolerant cycle
embedding of with both faulty vertices and faulty edges, and prove that
there exists a fault-free cycle of length in with
and for . Since is a
bipartite graph with two partite sets of equal size, the cycle of a length
is the longest in the worst-case.Comment: 17 pages, 5 figures, 1 tabl
On extremal cacti with respect to the edge Szeged index and edge-vertex Szeged index
The edge Szeged index and edge-vertex Szeged index of a graph are defined as
and
respectively, where
(resp., ) is the number of edges whose distance to
vertex (resp., ) is smaller than the distance to vertex (resp.,
), and (resp., ) is the number of vertices whose
distance to vertex (resp., ) is smaller than the distance to vertex
(resp., ), respectively. A cactus is a graph in which any two cycles have at
most one common vertex. In this paper, the lower bounds of edge Szeged index
and edge-vertex Szeged index for cacti with order and cycles are
determined, and all the graphs that achieve the lower bounds are identified.Comment: 12 pages, 5 figure
The generalized connectivity of -bubble-sort graphs
Let and denote the maximum number of
edge-disjoint trees in such that for any and . For an
integer with , the {\em generalized -connectivity} of a
graph is defined as and
. The generalized -connectivity is a generalization of the
traditional connectivity. In this paper, the generalized -connectivity of
the -bubble-sort graph is studied for . By
proposing an algorithm to construct internally disjoint paths in
, we show that for ,
which generalizes the known result about the bubble-sort graph [Applied
Mathematics and Computation 274 (2016) 41-46] given by Li , as the
bubble-sort graph is the special -bubble-sort graph for
On 3-extra connectivity of k-ary n-cubes
Given a graph G, a non-negative integer h and a set of vertices S, the
h-extra connectivity of G is the cardinality of a minimum set S such that G-S
is disconnected and each component of G-S has at least h+1 vertices. The
2-extra connectivity of k-ary n-cube is gotten by Hsieh et al. in [Theoretical
Computer Science, 443 (2012) 63-69]. In this paper, we obtained the h-extra
connectivity of the k-ary n-cube networks for h=3.Comment: This paper has been withdrawn by the author due to a crucial sign
error in the main theore
Relationship between Conditional Diagnosability and 2-extra Connectivity of Symmetric Graphs
The conditional diagnosability and the 2-extra connectivity are two important
parameters to measure ability of diagnosing faulty processors and
fault-tolerance in a multiprocessor system. The conditional diagnosability
of is the maximum number for which is conditionally
-diagnosable under the comparison model, while the 2-extra connectivity
of a graph is the minimum number for which there is a
vertex-cut with such that every component of has at least
vertices. A quite natural problem is what is the relationship between the
maximum and the minimum problem? This paper partially answer this problem by
proving for a regular graph with some acceptable
conditions. As applications, the conditional diagnosability and the 2-extra
connectivity are determined for some well-known classes of vertex-transitive
graphs, including, star graphs, -star graphs, alternating group
networks, -arrangement graphs, alternating group graphs, Cayley graphs
obtained from transposition generating trees, bubble-sort graphs, -ary
-cube networks and dual-cubes. Furthermore, many known results about these
networks are obtained directly
The Component Connectivity of Alternating Group Graphs and Split-Stars
For an integer , the -component connectivity of a
graph , denoted by , is the minimum number of vertices
whose removal from results in a disconnected graph with at least
components or a graph with fewer than vertices. This is a natural
generalization of the classical connectivity of graphs defined in term of the
minimum vertex-cut and is a good measure of robustness for the graph
corresponding to a network. So far, the exact values of -connectivity are
known only for a few classes of networks and small 's. It has been
pointed out in~[Component connectivity of the hypercubes, Int. J. Comput. Math.
89 (2012) 137--145] that determining -connectivity is still unsolved for
most interconnection networks, such as alternating group graphs and star
graphs. In this paper, by exploring the combinatorial properties and
fault-tolerance of the alternating group graphs and a variation of the
star graphs called split-stars , we study their -component
connectivities. We obtain the following results: (i) and
for , and for
; (ii) , , and
for
- β¦