Abstract 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 WE = {w1, w2, . . ., wk} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e, w1), d(e, w2), . . ., d(e, wk)). If distinct edges of G have distinct representation with respect to WE, then WE 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 Cn(1, m) has vertex set {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn(1, 2) and Cn(1, 3) is constant.

中文翻译：

---

某些类循环图的边度量维数

摘要 令 G = (V (G), E(G)) 为连通图且 x, y ∈ V (G), d(x, y) = min{ x − y path 的长度 } 且对于 e ∈ E (G), d(x, e) = min{d(x, a), d(x, b)}，其中 e = ab。如果 d(e1, x) ≠ d(e2, x)，则顶点 x 区分两条边 e1 和 e2。令 WE = {w1, w2, ... . ., wk} 是 V (G) 中的有序集，令 e ∈ E(G)。e 相对于 WE 的表示 r(e | WE) 是 k 元组 (d(e, w1), d(e, w2), ..., d(e, wk))。如果 G 的不同边相对于 WE 具有不同的表示，则 WE 称为 G 的边度量生成器。 最小基数的边度量生成器是 G 的边度量基，其基数称为 G 的边度量维数，由 edim(G) 表示。循环图 Cn(1, m) 的顶点集 {v1, v2, ... . ., vn} 和边集 {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤米}。