Let Γ 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 graph Γ is a subset of vertices Z ⊆ V (Γ) such that DOI:http://dx.doi.org/10.29252/asta.4.2.15 MSC(2010): Primary: 05C25. Secondary: 05C75.

Let Γ be a graph on n vertices. A subset W of V (Γ) is called a resolving set when for each u, v ∈ V (Γ) there exists w ∈ W such that ∂(u, w) = ∂(v, w). The metric dimension of Γ is the minimum cardinality among resolving sets of Γ and is denoted by dim(Γ). This parameter has many applications in chemistry, in the navigation of robots in networks and in the problems of pattern recognition and image processing some of which involve the use of hierarchical data structures. In this paper, we study the metric dimension of Cayley graphs. Specially, we present a complete characterization of Cayley graphs on Abelian groups whose metric dimension is two.

A subset W of the vertices of a graph G is a resolving set for G when for each (v, w). The cardinality of a minimum resolving set for G is the metric dimension of G. This concept has applications in many diverse areas including network discovery, robot navigation, image processing, combinatorial search and optimization. The problem of finding metric dimension is NP-complete for general graphs but the metric dimension of trees can be obtained using a polynomial time algorithm. In this paper, we investigate the metric dimension of Cayley graphs on dihedral groups and we characterize a family of them. x is N (x) = {y : x ∼ y}. A walk consists of an alternating sequence of vertices and edges http://dx.doi.org/10.29252/asta.4.1.43 MSC(2010): Primary: 05C25. Secondary: 05C75, 20B10.

One of the most interesting problems on maximum nullity (minimum rank) is to characterize M (G) (mr(G)) for a graph G. In this regard, many researchers have been trying to find an upper or lower bound for the maximum nullity. For more results on this topic, see [6], [7], [11] and [12]. In this paper, by using a result of Babai [5], which presents the spectrum of a Cayley graph in terms of irreducible characters of the underlying group, and using representation and character of groups, we give a lower bound for the maximum nullity of Cayley graph, X S (G), where G = a is a cyclic group, or G = G 1 × · · · × G t such that G 1 = a is a cyclic group and G i is an arbitrary finite group, for some 2 ≤ i ≤ t, with determine the spectrum of Cayley graphs.