81 research outputs found
Distinguishing number and distinguishing index of natural and fractional powers of graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. For any , the -subdivision of is a simple graph
which is constructed by replacing each edge of with a path of length .
The power of , is a graph with same set of vertices of and an
edge between two vertices if and only if there is a path of length at most
between them. The fractional power of , denoted by is
power of the -subdivision of or -subdivision of -th power
of . In this paper we study the distinguishing number and distinguishing
index of natural and fractional powers of . We show that the natural powers
more than two of a graph distinguished by three edge labels. Also we show that
for a connected graph of order with maximum degree , and for , .Comment: 13 page
Distinguishing number and distinguishing index of neighbourhood corona of two graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. The neighbourhood
corona of two graphs and is denoted by and is the
graph obtained by taking one copy of and copies of , and
joining the neighbours of the th vertex of to every vertex in the
th copy of . In this paper we describe the automorphisms of the graph
. Using results on automorphisms, we study the distinguishing
number and the distinguishing index of . We obtain upper bounds
for and .Comment: 15 pages, 11 figure
Distinguishing number and distinguishing index of graphs from primary subgraphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. Let be a connected
graph constructed from pairwise disjoint connected graphs by
selecting a vertex of , a vertex of , and identify these two
vertices. Then continue in this manner inductively. We say that is obtained
by point-attaching from and that 's are the primary
subgraphs of . In this paper, we consider some particular cases of these
graphs that are of importance in chemistry and study their distinguishing
number and index.Comment: 15 pages, 13 figure
The distinguishing number and the distinguishing index of line and graphoidal graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. A graphoidal cover of
is a collection of (not necessarily open) paths in such that
every path in has at least two vertices, every vertex of is an
internal vertex of at most one path in and every edge of is in
exactly one path in . Let denote the intersection graph
of . A graph is called a graphoidal graph, if there exists a graph
and a graphoidal cover of such that .
In this paper, we study the distinguishing number and the distinguishing index
of the line graph and the graphoidal graph of a simple connected graph .Comment: 9 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1707.0616
Distinguishing number and distinguishing index of certain graphs
The distinguishing number (index) () of a graph is the
least integer such that has an vertex labeling (edge labeling) with
labels that is preserved only by a trivial automorphism. In this paper we
compute these two parameters for some specific graphs. Also we study the
distinguishing number and the distinguishing index of corona product of two
graphs.Comment: 15 pages, 6 figures....To appear in FILOMA
Relationship between the distinguishing index, minimum degree and maximum degree of graphs
Let and be the minimum and the maximum degree of the
vertices of a simple connected graph , respectively.
The distinguishing index of a graph , denoted by , is the least
number of labels in an edge labeling of not preserved by any non-trivial
automorphism. Motivated by a conjecture by Pil\'sniak (2017) that implies that
for any -connected graph , we
prove that for any graph with , . Also, we show that the distinguishing index
of -regular graphs is at most , for any .Comment: 8 pages. arXiv admin note: substantial text overlap with
arXiv:1702.03524, arXiv:1704.0415
Characterization of graphs with distinguishing number equal list distinguishing number
The distinguishing number of a graph is the least integer such
that has an vertex labeling with labels that is preserved only by a
trivial automorphism. A list assignment to is an assignment of lists of labels to the vertices of . A
distinguishing -labeling of is a distinguishing labeling of where
the label of each vertex comes from . The list distinguishing number
of , is the minimum such that every list assignment to in
which for all yields a distinguishing -labeling
of . In this paper, we determine the list-distinguishing number for two
families of graphs. We also characterize graphs with the distinguishing number
equal the list distinguishing number. Finally, we show that this
characterization works for other list numbers of a graph.Comment: 10 pages, 2 figure
The chromatic distinguishing index of certain graphs
The distinguishing index of a graph , denoted by , is the least
number of labels in an edge labeling of not preserved by any non-trivial
automorphism. The distinguishing chromatic index of a graph
is the least number such that has a proper edge labeling with
labels that is preserved only by the identity automorphism of . In this
paper we compute the distinguishing chromatic index for some specific graphs.
Also we study the distinguishing chromatic index of corona product and join of
two graphs.Comment: 12 pages, 2 figure
The distinguishing chromatic number of bipartite graphs of girth at least six
The distinguishing number of a graph is the least integer such
that has a vertex labeling with labels that is preserved only by a
trivial automorphism. The distinguishing chromatic number of
is defined similarly, where, in addition, is assumed to be a proper
labeling. Motivated by a conjecture in \cite{colins}, we prove that if is a
bipartite graph of girth at least six with the maximum degree ,
then . We also obtain an upper bound for
where is a graph with at most one cycle. Finally, we state a
relationship between the distinguishing chromatic number of a graph and its
spanning subgraphs.Comment: 6 page
The cost number and the determining number of a graph
The distinguishing number of a graph is the least integer such
that has an vertex labeling with labels that is preserved only by a
trivial automorphism. The minimum size of a label class in such a labeling of
with is called the cost of -distinguishing and is denoted
by . A set of vertices is a determining set for
if every automorphism of is uniquely determined by its action on .
The determining number of , Det(G), is the minimum cardinality of
determining sets of .
In this paper we obtain some general upper and lower bounds for
based on Det(G). Finally, we compute the cost and the determining number for
the friendship graphs and corona product of two graphs.Comment: 8 page
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