arXiv subjects

Saeid Alikhani

Publications and source records attributed to Saeid Alikhani.

At least 91 records · Page 5Linked to original sources

arXiv2017

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 $D'(G) \leq D(G)$.

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arXiv2017

More on the sixth coefficient of the matching polynomial in regular graphs

A matching set $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. In this paper we consider some parameters related to the matching of regular graphs. We find the sixth coefficient of the matching polynomial of regular graphs. As a consequence, every cubic graph of order $10$ is matching unique.

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arXiv2017

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 $ρ_d(G)$. 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 $ρ_d(G)$ based on Det(G). Finally, we compute the cost and the determining number for the friendship graphs and corona product of two graphs.

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arXiv2017

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 $χ_{D}(G)$ 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 $Δ(G)$, then $χ_{D}(G)\leq Δ(G)+1$. We also obtain an upper bound for $χ_{D}(G)$ 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.

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arXiv2017

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 $ψ$ of (not necessarily open) paths in $G$ such that every path in $ψ$ has at least two vertices, every vertex of $G$ is an internal vertex of at most one path in $ψ$ and every edge of $G$ is in exactly one path in $ψ$. Let $Ω(G,ψ)$ denote the intersection graph of $ψ$. A graph $H$ is called a graphoidal graph, if there exists a graph $G$ and a graphoidal cover $ψ$ of $G$ such that $H\cong Ω(G, ψ)$. 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 $G$.

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arXiv2017

Trees with distinguishing index equal distinguishing number plus one

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 $D'(G) \leq D(G) + 1$. In this note we characterize trees for which this inequality is sharp. We also show that if $G$ is a connected unicyclic graph, then $D'(G) = D(G)$.

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arXiv2017

The distinguishing number (index) and the domination number of a graph

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 set $S$ of vertices in $G$ is a dominating set of $G$ if every vertex of $V(G)\setminus S$ is adjacent to some vertex in $S$. The minimum cardinality of a dominating set of $G$ is the domination number of $G$ and denoted by $γ(G)$. In this paper, we obtain some upper bounds for the distinguishing number and the distinguishing index of a graph based on its domination number.

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arXiv2017

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.

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arXiv2017

More on the total dominator chromatic number of a graph

Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. 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 study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of $r$-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the TDC-number of $G$. We study the stability and bondage number of certatin graphs.

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arXiv2017

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

Let $δ$ and $Δ$ 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 Pilśniak (2017) that implies that for any $2$-connected graph $D'(G) \leq \lceil \sqrt{Δ(G)}\rceil +1$, we prove that for any graph $G$ with $δ\geq 2$, $D'(G) \leq \lceil \sqrt[δ]{Δ}\rceil +1$. Also, we show that the distinguishing index of $k$-regular graphs is at most $2$, for any $k\geq 5$.

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arXiv2017

Some new results on the total domination polynomial of a graph

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $γ_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=γ_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. A root of $D_t(G,x)$ is called a total domination root of $G$. An irrelevant edge of $D_t(G,x)$ is an edge $e \in E$, such that $D_t(G, x) = D_t(G\setminus e, x)$. In this paper, we characterize edges possessing this property. Also we obtain some results for the number of total dominating sets of a regular graph. Finally, we study graphs with exactly two total domination roots $\{-3,0\}$, $\{-2,0\}$ and $\{-1,0\}$.

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arXiv2017

The distinguishing number and the distinguishing index of Cayley graphs

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arXiv2017

Distinguishing number and distinguishing index of strong 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 strong product $G\boxtimes H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(x_1, x_2), (y_1, y_2)\} | x_iy_i \in E(G_i) ~{\rm or}~ x_i = y_i ~{\rm for~ each}~ 1 \leq i \leq 2.\}$. In this paper we study the distinguishing number and the distinguishing index of strong product of two graphs. We prove that for every $k \geq 2$, the $k$-th strong power of a connected $S$-thin graph $G$ has distinguishing index equal 2.

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arXiv2017

Counting the number of weakly connected dominating sets of graphs

Let $G=(V(G),E(G))$ be a simple graph. A non-empty set $S\subseteq V (G)$ is a weakly connected dominating set in $G$, if the subgraph obtained from $G$ by removing all edges each joining any two vertices in $V (G)\setminus S$ is connected. In this paper, we consider some graphs and study the number of their weakly connected dominating sets.

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arXiv2017

An upper bound on the distinguishing index of graphs with minimum degree at least two

The distinguishing index of a simple 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. It was conjectured by Pilśniak (2015) that for any 2-connected graph $D'(G) \leq \lceil \sqrt{Δ(G)}\rceil +1$. We prove a more general result for the distinguishing index of graphs with minimum degree at least two from which the conjecture follows. Also we present graphs $G$ for which $D'(G)\leq \lceil \sqrt{Δ}\rceil$.

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arXiv2017

Distinguishing number and distinguishing index of join 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. In this paper we study the distinguishing number and the distinguishing index of join of two graphs $G$ and $H$, i.e., $G+H$. We prove that $0\leq D(G+H)-max\{D(G),D(H)\}\leq z$, where $z$ is depends of the number of some induced subgraphs generated by some suitable partitions of $V(G)$ and $V(H)$. Also, we prove that if $G$ is a connected graph of order $n \geq 2$, then $D'(G+ \cdots +G)=2$, except $D'(K_2+K_2)=3$.

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arXiv2016

The distinguishing number of groups based on the distinguishing number of subgroups

Let $Γ$ be a group acting on a set $X$. The distinguishing number for this action of $Γ$ on $X$, denoted by $D_Γ(X)$, is the smallest natural number $k$ such that the elements of $X$ can be labeled with $k$ labels so that any label-preserving element of $Γ$ fixes all $x \in X$. In particular, if the action is faithful, then the only element of $Γ$ preserving labels is the identity. In this paper, we obtain an upper bound on the distinguishing number of a set knowing the distinguishing number of a set under the action of a subgroup. By the concept of motion, we obtain an upper bound for the distinguishing number of a group. Motivated by a problem (Chan 2006), we characterize $D_{Γ,H}(X)$ which is the smallest number of labels admitting a labeling of $X$ such that the only elements of $Γ$ that induce label-preserving permutations lie in $H$. Finally, we state two algorithms for obtaining an upper and a lower bound for $D_{Γ, H}(X)$.

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arXiv2016

Trees with distinguishing number two

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. In this paper we characterize all trees with radius at most three and distinguishing number two. Also we present a necessary condition for trees with distinguishing number two and radius more than three.

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