On the zero forcing number of some Cayley graphs

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran

2 Department of Mathematics, Faculty of sciences, Imam Khomeini International University, Qazvin, Iran

10.29252/asta.4.2.15

Abstract

‎Let Γa be a graph whose each vertex is colored either white or black‎. ‎If u is a black vertex of Γ such that exactly one neighbor‎ ‎v of u is white‎, ‎then u changes the color of v to black‎. ‎A zero forcing set for a Γ graph is a subset of vertices Z\subseteq V(Γ) such that‎ if initially the vertices in Z are colored black and the remaining vertices are colored white‎, ‎then Z changes the color of all vertices Γ in to black‎. ‎The zero forcing number of Γ is the minimum of |Z| over all zero forcing sets for Γ and is denoted by Z(Γ)‎. In this paper‎, ‎we consider the zero forcing number of some families of Cayley graphs‎. ‎In this regard‎, ‎we show that Z(Cay(D2n,S))=2|S|-2‎, ‎where D2n is dihedral group of order 2n and S={a‎, ‎a3‎, ‎... ‎, ‎a2k-1‎, ‎b}. ‎Also‎, ‎we obtain Z(Cay(G,S))‎, ‎where G=< a> is a cyclic group of even order n and S={ai :‎ 1≤ i≤ n‎ and i is odd}‎, ‎S={ai‎ :‎1≤ i≤ n‎ and i is odd}\{ak,a-k} or |S|=3‎.

Keywords


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