On restricted edge-connectivity of lexicographic product graphs

Let G1 be a nontrivial connected graph and G2 be a nonempty graph. Denote by G1° G2 the lexicographic product of G1 and G2. Let λ′(G1° G2) and ξ(G1° G2) represent the restricted edge-connectivity and the minimum edge-degree of G1° G2, respectively. In this paper, we show that , where ni, λi, δi and ξi are the order, the edge-connectivity, the minimum degree and the minimum edge-degree of Gi (i=1, 2) respectively. Moreover, when the order of G2 is at least five, we prove that: (i) G1° G2 is super restricted edge-connected if and only if ; (ii) if G1 is maximally edge-connected, then G1° G2 is super restricted edge-connected; (iii) if G1 is maximally restricted edge-connected, then G1° G2 is super restricted edge-connected.