The Distinguishing Number of Kronecker 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 an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. The Kronecker product G × H of two graphs G and H is the graph with vertex set V (G) × V (H) and edge set {{(u, x), (v, y)}|{u, v} ∈ E(G) and {x, y} ∈ E(H)}. In this paper we study the distinguishing number and the distinguishing index of Kronecker product of two graphs.

‎Distinguishing index of Kronecker product of two graphs

“…We begin with the distinguishing number of Kronecker product of complete graphs. The results of this section has presented in the "International conference on theoretical computer science and discrete mathematics, ICTCSDM 2016" and has published by authors in its proceeding [2]. Because of this, we just state the results and for the proof, readers can see [2].…”

“…Proposition 2.1. [2] If K m,n and K p,q are complete bipartite graphs such that q p and m n then the distinguishing number of K m,n × K p,q is…”

“…Theorem 2.5. [2] If G and H are two simple connected, relatively prime graphs such that G × H has j i R-equivalence classes of sie a i for i = 1, . .…”

“…The distinguishing index of some examples of graphs was exhibited in [15]. The distinguishing number and the distinguishing index of some graph products has been studied in literature (see [2,3,4,12,13]). The Cartesian product of graphs G and H is a graph, denoted G2H, whose vertex set is V (G) × V (H).…”

“…Weichsel [20] proved that the Kronecker product of two nontrivial graphs is connected if and only if both factors are connected and at least one of them possesses an odd cycle. If both factors are connected and bipartite, then their Kronecker product consists of two connected components ( [2]). The Kronecker product G × H of graphs G and H is the graph with vertex set V (G) × V (H) and edge set {{(u, x), (v, y)}|{u, v} ∈ E(G) and {x, y} ∈ E(H)}.…”