1. Extra Edge Connectivity and Isoperimetric Edge Connectivity Zhang Zhao College of Mathematics and System Sciences Xinjiang University

2. Backgroud: Network Reliability Classical measure: λ(G) Generalization of edge connectivity:  Restricted edge connectivity (Extra edge connectivity)  Isoperimetric edge connectivity

3. Restricted edge connectivity Let G be a connected graph and S an edge subset. If G – S is disconnected and each component of G – S has at least k vertices, then S is said to be a k-restricted edge cut. The cardinality of a minimum k-restricted edge cut is called the k- restricted edge connectivity of G, denoted by λ k (G).

4. Restricted edge connectivity Graphs having k-restricted edge cut is said to be λ k -connected. The larger λ k is, the more reliable the network is.

5. Isoperimetric edge connectivity Remark. λ k can be rewritten as both connected} (Hamidoune 2000) The k-th isoperimetric edge connectivity is defined as

6. Question: When does γ k coincide with λ k ? (Hamidoune 2000) γ k coincide with λ k in undirected graphs. This is wrong. (Wang & Li 2002) γ k = λ k when G is a d-regular connected edge-transitive graph with d ≥ 3k.

7. Theorem. Let G be a connected d-regular graph of order n and girth g. If G is λ k -connected and k ≤ 2g, then γ k (G) = λ k (G). Remark. When k ≤ 6, γ k (G) = λ k (G) for any regular graph G with order at least 2k.

9. γ k -optimal graphs Define Then γ k (G) ≤ β k (G). A graph G is said to be γ k -optimal if γ k (G) = β k (G)

10. γ k -optimal graphs (Zhang & Huang 2004) Let G be a connected d- regular graph with d ≥ 4. If G is vertex-transitive with girth 5 ≤ g ≤ [|V(G)|/2], or G is edge- transitive with 4 ≤ g ≤ [|V(G)|/2], then G is γ g - optimal. Remark. Under the above assumption, G is also λ g - optimal.

11. γ k -optimal graphs Theorem. Let G be a 3-regular connected vertex- transitive or edge-transitive graph with girth g. Then G is γ k -optimal if and only if g ≥ k+2.

12. γ k -optimal graphs Theorem. Let G be a connected d-regular edge- transitive graph with d ≥ 6e k (G)/k. Then G is γ k - optimal. Where e k (G) is the minimum number of edges in an induced subgraph of G with order k. (Hamidoune 2000) A connected d-regular edge-transitive graph G with d ≥ 3(k – 1) is γ k - optimal.