Edge metric dimension of some classes of circulant graphs

Let G = (V (G), E (G)) be a connected graph and x, y is an element of V(G), d(x , y) = min{ length of x - y path } and for e is an element of E(G), d(x,e) = min{d(x, a), d(x , b)} , where e = ab. A vertex x distinguishes two edges e(1) and e(2), if d(e(1), x) not equal d(e(2), x). Let W-E = {w(1),w(2), ...,w(k)} be an ordered set in V(G) and let e is an element of E(G). The representation r(e vertical bar W-E) of e with respect to W-E is the k-tuple (d (e, w(1)) , d(e, w(2)) , ..., 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 {v(1), v(2), ..., v(n)} and edge set {v(i)v(i+1) : 1 <= i <= n - 1}boolean OR{v(n)v(1)}boolean OR{v(i)v(i+m) : 1 <= i <= n - m}boolean OR(v(n-m+i)v(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.