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... 1 divides n, and a related construction in other cases. 1 Introduction In this paper, we are concerned with finding an upper bound on the metric dimension of Grassmann graphs. We use the notation of =-=[3]-=-. To define the metric dimension of a graph, we need the following concept. Definition 1. A resolving set for a graph Γ = (V,E) is a set of vertices S = {v1,...,vk} such that for all w ∈ V , the list ...

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...ng number or rigidity problem. The problem was defined independently by Harary and Melter [12] and Slater [18], and has received a lot of attention from researchers in different disciplines (see e.g. =-=[2,13]-=- and references therein). For example, a recent survey by Bailey and Cameron [2] notes an interesting connection to group theory and graph isomorphism. Computing the metric dimension has proved to be ...

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...eometry [9], robot navigation, coin weighing problems, connected joins in graphs, and strategies for the Mastermind game. We refer to Cáceres et al. [4], Hernando et al. [11], and Bailey and Cameron =-=[2]-=- for a more comprehensive list and a more complete bibliography. From a computational complexity point of view, Metric Dimension is known to be NP-hard and there is a linear-time algorithm for trees [...

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...to have a regular orbit on the power set of the vertex set (see [2, Proposition 2.7]) and partly because this property tends to be generic for a variety of combinatorial and algebraic structures (see =-=[2]-=- and [6], for example). Some classes of infinite graphs known to have distinguishing number 2 include all locally finite trees having no 1-valent vertex [13]; the infinite hypercube of dimension n, th...

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...ng on the vertex set and extended to general group actions by Tymoczko [25] (see also [4, 5, 27]). On the other hand, similar ideas are found earlier in permutation groups; indeed, Bailey and Cameron =-=[3]-=- cite many situations where graph theorists have the electronic journal of combinatorics 18 (2011), #P50 1rediscovered and renamed concepts from permutation groups. The graph theoretic origin of dist...

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...e [16, 17]. For more on zero forcing parameter in graphs, see [3, 4, 12, 13, 15, 22]. More recently, the comparative study of graph parameters is becoming – it appears – increasingly fashionable; see =-=[2]-=-, [5], and [9], for examples. Our work here is inspired, in part, by these comparative studies. It is also inspired by our observation of the coincidence between zero forcing number and metric dimensi...

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...ations of closely-related parameters were given by Babai in the study of the graph isomorphism problem [1] and in obtaining bounds on the possible orders of primitive permutation groups [2] (see also =-=[3]-=-). Since the problem of computing the metric dimension of a graph is NP-complete (see [26]), many efforts have been focused on finding either exact values or good bounds for the metric dimension of ce...

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...Further, [n] next to a formula indicates that the formula can be found in reference [n]. L(G) Metric Dimension Zero Forcing Number L(T ), T 6= Pn σ(T )− ex(T ) [15] σ(T )− 1 [1] L(Kn), n ≥ 6 ⌈ 2n 3 ⌉ =-=[2]-=- 1 2 (n 2 − 3n+ 4) [1] L(Ks,t), s, t ≥ 2 { ⌊ 2(s+t−1)3 ⌋ if s ≤ t ≤ 2s− 1 t− 1 if t ≥ 2s [6] st− s− t+ 2 [1] L(W1,n), n ≥ 3 3 if n = 3, 4 4 if n = 5 n− ⌈n3 ⌉ if n ≥ 6 n+ 1 L(Bn), n ≥ 2 2n− 1 2n− ...