Edge metric dimension of some classes of circulant graphs

Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e₁ and e₂, if d(e₁, x) ≠ d(e₂, x). Let W_E = {w₁, w₂, . . ., w_k} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | W_E) of e with respect to W_E is the k-tuple (d(e, w₁), d(e, w₂), . . ., d(e, w_k)). If distinct edges of G have distinct representation with respect to W_E, then W_E is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph C_n(1, m) has vertex set {v₁, v₂, . . ., v_n} and edge set {v_iv_i+1 : 1 ≤ i ≤ n−1}∪{v_nv₁}∪{v_iv_i+m : 1 ≤ i ≤ n−m}∪{v_n−m+iv_i : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs C_n(1, 2) and C_n(1, 3) is constant.