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 $D'(K_2+K_2)=3$.
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing edge-colouring with two colours. We show that all such graphs except $K_2$ admit a distinguishing edge-colouring with three colours. This result also extends to infinite, locally finite graphs. Furthermore, we are able to show that there are arbitrary large infinite cardinals $κ$ such that every connected $κ$-regular graph has distinguishing edge-colouring with two colo…
An edge-coloring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Lehner, Pilśniak, and Stawiski proved that all connected regular graphs except $K_2$ admit an asymmetric edge-coloring with three colors. We generalize this result for graphs whose minimal degree $δ$ and the maximal degree $Δ$ satisfy $δ\geq Δ/2$.
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{…
The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $α$-regular graph for some infinite cardinal $α$ then $D'(G) \le 2$, proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if $G$ is a graph with infinite minimum degree and at most $2^α$ vertices of degree $α$ for every infinite cardinal $α$, then $D'(G) \le 3$. In particular, $D'(G) \le 3$ if $G$ has infinite minimum degree and order at most $2^{\aleph_0}$.
We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $χ_D(G)$, and the distinguishing chromatic index $χ'_D(G)$, for vertex and edge colourings, respectively. We show that $χ_D(G)\leq 2Δ(G)-1$ for graphs with a finite maximum degree $…
In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring $γ$ of a digraph $D$ is proper if no two arcs with the same head or with the same tail are assigned the same colour. For each vertex $u$ of $D$, we denote by $S_γ^-(u)$ and $S_γ^+(u)$ the sets of colours that appear on the incoming arcs and on the outgoing arcs of $u$, respectively. An arc colouring $γ$ of $D$ is \emph{neighbour-distinguishing} if, for every two adjacent vertices $u$ and $v$ of $D$, the ordered pairs $(S_γ^-(u),S_γ^+(u))$ and $(S_γ^-(v),S_γ^+(v))$ are dis…
In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice $L$ and its set $Q_L$ of join-irreducibles, we use classic lattice theory to show that any linear extension of $Q_L$ generates a distinguishing 2-coloring of $L$. We prove general upper bounds for the distinguishing chromatic number and particular upper bounds for the Boolean lattice and for divisibility lattices. In addition, we show that the distinguishing number of any twin-free Cohen-Macaulay planar lattice is at most 2.
A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping $λ$ from the set of vertices (resp. the set of edges) of G to the set of labels {1,. .. , r} such that no non-trivial automorphism of G preserves all the vertex (resp. edge) labels. The distinguishing number D(G) and the distinguishing index D (G) of G are then the smallest r for which G admits a distinguishing r-vertex-labelling or r-edge-labelling, respectively. The distinguishing chromatic number D $χ$ (G) and the distinguishing chromatic index D $χ$ (G) are defined similarly, with the additio…
A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a coloring. This paper is about counting non-equivalent distinguishing colorings of graphs with $k$ colors. A parameter, namely $Φ_k (G)$, which is the number of non-equivalent distinguishing colorings of a graph $G$ with at most $k$ colors, is shown here to have an application in calculating the distinguishing number of the lexicographic product and the $X$-join o…
The distinguishing index $D'(Γ)$ of a graph $Γ$ is the least number $k$ such that $Γ$ has an edge-coloring with $k$ colors preserved only by the trivial automorphism. In this paper we prove that if the automorphism group of a finite graph $Γ$ is simple, then its distinguishing index $D'(Γ)=2$.
Topological indices are molecular descriptors that describe the properties of chemical compounds. These topological indices correlate specific physico-chemical properties like boiling point, enthalpy of vaporization, strain energy, and stability of chemical compounds. This article introduces a new graph operation based on central graph called central vertex-edge join and provides its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmet…
The union vertex-distinguishing chromatic index $χ'_\cup(G)$ of a graph $G$ is the smallest natural number $k$ such that the edges of $G$ can be assigned nonempty subsets of $[k]$ so that the union of the subsets assigned to the edges incident to each vertex is different. We prove that $χ'_\cup(G) \in \left\{ \left\lceil \log_2\left(n +1\right) \right\rceil, \left\lceil \log_2\left(n +1\right) \right\rceil+1 \right\}$ for a graph $G$ on $n$ vertices without a component of order at most two. This answers a question posed by Bousquet, Dailly, Duchêne, Kheddouci and Parreau, and independently by …
A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph $G$, denoted by $θ(G)$, is the minimum number of colors $k$ so that every $k$-coloring of $G$ is distinguishing. We generalize this concept to edge-coloring by defining an alternative index $θ'(G)$. We consider $θ'$ for some families of graphs and find its connection with edge-cycles of the automorphism group. Then we show that $θ'(G)=2$ if and only if $G\simeq K_{1,2}$ and $θ'(G)=3$ if and only if $G\simeq P_4, K_{1,3}$ or $K_3$. Moreover, we prove some aux…
Very recently, a bond-additive topological descriptor, known as the Mostar index, has been proposed as a measure of peripherality in graphs and networks. In this article, we compute the Mostar index of corona product, Cartesian product, join, lexicographic product, Indu-Bala product and subdivision vertex-edge join of graphs and apply these results to find the Mostar index of various classes of chemical graphs and nanostructures.
In this paper, we compute normalized Laplacian spectra of central graph of a regular graph, central vertex join and central edge join of two regular graphs. Also, we determine their Kemeny's constant and degree Kirchhoff index.
