Distinguishing number and distinguishing index of natural and fractional powers of graphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. For any $n \in \mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$. The $m^{th}$ power of $G$, is a graph with same set of vertices of $G$ and an edge between two vertices if and only if there is a path of length at most $m$ between them. The fractional power of $G$, denoted by $G^{\frac{m}{n}}$ is $m^{th}$ power of the $n$-subdivision of $G$ or $n$-subdivision of $m$-th power of $G$. In this paper we study the distinguishing number and distinguishing index of natural and fractional powers of $G$. 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 $G$ of order $n \geqslant 3$ with maximum degree $\Delta (G)$, $D(G^{\frac{1}{k}})\leqslant min\{s: 2^k+\sum^s_{n=3}n^{k-1}\geqslant \Delta (G)\}$ and for $m\geqslant 3$, $D'(G^{\frac{m}{k}})\leqslant 3$.Comment: 13 page

Distinguishing number and distinguishing index of neighbourhood corona of two graphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on automorphisms, we study the distinguishing number and the distinguishing index of $G_1\star G_2$. We obtain upper bounds for $D(G_1\star G_2)$ and $D'(G_1\star G_2)$.Comment: 15 pages, 11 figure

Distinguishing number and distinguishing index of graphs from primary subgraphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$. 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) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A graphoidal cover of $G$ is a collection $\psi$ of (not necessarily open) paths in $G$ such that every path in $\psi$ has at least two vertices, every vertex of $G$ is an internal vertex of at most one path in $\psi$ and every edge of $G$ is in exactly one path in $\psi$. Let $\Omega(G,\psi)$ denote the intersection graph of $\psi$. A graph $H$ is called a graphoidal graph, if there exists a graph $G$ and a graphoidal cover $\psi$ of $G$ such that $H\cong \Omega (G, \psi)$. 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 $G$.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) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ 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

The cost number and the determining number of a graph

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The minimum size of a label class in such a labeling of $G$ with $D(G) = d$ is called the cost of $d$-distinguishing $G$ and is denoted by $\rho_d(G)$. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of $G$, Det(G), is the minimum cardinality of determining sets of $G$. In this paper we obtain some general upper and lower bounds for $\rho_d(G)$ 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

The distinguishing chromatic number of bipartite graphs of girth at least six

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $\chi_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. Motivated by a conjecture in \cite{colins}, we prove that if $G$ is a bipartite graph of girth at least six with the maximum degree $\Delta (G)$, then $\chi_{D}(G)\leq \Delta (G)+1$. We also obtain an upper bound for $\chi_{D}(G)$ where $G$ 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 distinguishing index of graphs with at least one cycle is not more than its distinguishing number

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$ we have $D'(G) \leq D(G) + 1$. The complete characterization of finite trees $T$ with $D'(T)=D(T)+ 1$ has been given recently. In this note we show that if $G$ is a finite connected graph with at least one cycle, then $D'(G)\leq D(G)$. Finally, we characterize all connected graphs for which $D'(G) \leq D(G)$

The distinguishing number and the distinguishing index of co-normal product of two graphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The co-normal product $G\star H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(x_1, x_2), (y_1, y_2)\} | x_1y_1 \in E(G) ~{\rm or}~x_2y_2 \in E(H)\}$. In this paper we study the distinguishing number and the distinguishing index of the co-normal product of two graphs. We prove that for every $k \geq 3$, the $k$-th co-normal power of a connected graph $G$ with no false twin vertex and no dominating vertex, has the distinguishing number and the distinguishing index equal two.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1703.0187

Relationship between the distinguishing index, minimum degree and maximum degree of graphs

Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. Motivated by a conjecture by Pil\'sniak (2017) that implies that for any $2$-connected graph $D'(G) \leq \lceil \sqrt{\Delta (G)}\rceil +1$, we prove that for any graph $G$ with $\delta\geq 2$, $D'(G) \leq \lceil \sqrt[\delta]{\Delta }\rceil +1$. Also, we show that the distinguishing index of $k$-regular graphs is at most $2$, for any $k\geq 5$.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1702.03524, arXiv:1704.0415