Samaneh Soltani

Distinguishing number and distinguishing index of some operations on graphs

Abstract The distinguishing number (index) D(G) (Dʹ(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. We examine the effects on D(G) and Dʹ(G) when G is modified by operations on vertex and edge of G. Let G be a connected graph of order n ≥ 3. We show that –1 ≤ D(G – v) – D(G) ≤ D(G), where G – v denotes the graph obtained from G by removal of a vertex v and all edges incident to v and these inequalities are true for the distinguishing index. Also we prove that |D(G – e) – D(G)| ≤ 2 and –1 ≤ Dʹ (G – e) – Dʹ(G) ≤ 2, where G – e denotes the graph obtained from G by simply removing the edge e. Finally we consider the vertex contraction and the edge contraction of G and prove that the edge contraction decrease the distinguishing number (index) of G by at most one and increase by at most 3D(G) (3Dʹ(G)).

The distinguishing number and distinguishing index of the lexicographic product of two graphs

Abstract The distinguishing number (index) D(G) (D′(G)) of a graph G is the least integer d such that G has a vertex labeling (edge labeling) with d labels that is preserved only by the trivial automorphism. The lexicographic product of two graphs G and H, G[H] can be obtained from G by substituting a copy Hu of H for every vertex u of G and then joining all vertices of Hu with all vertices of Hv if uv ∈ E(G). In this paper we obtain some sharp bounds for the distinguishing number and the distinguishing index of the lexicographic product of two graphs. As consequences, we prove that if G is a connected graph with Aut(G[G]) = Aut(G)[Aut(G)], then for every natural number k, D(G) ≤ D(Gk) ≤ D(G) + k − 1 and all lexicographic powers of G, Gk (k ≥ 2) can be distinguished by two edge labels, where Gk = G[G[. . . ]].

Trees with distinguishing index equal distinguishing number plus one

Abstract 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) ≤ D(G) + 1. In this note we characterize finite trees for which this inequality is sharp. We also show that if G is a connected unicyclic graph, then D′ (G) = D(G).

The Distinguishing Number of Kronecker Product of Two Graphs

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 Kronecker product \(G\times H\) of two graphs G and H is the graph with vertex set \(V (G)\times V (H)\) and edge set \(\{\{(u, x), (v, y)\} | \{u, v\} \in E(G) ~and ~\{x, y\} \in E(H)\}\). In this paper we study the distinguishing number of Kronecker product of two graphs.