For a connected graph the restricted edge-connectivity λ (G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in (2004), 113-120] gave a sufficient condition for λ -optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g − 1, g being the girth of the graph, and show that a graph G with diameter at most g − 2 is λ -optimal.

The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters ( and , edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G m ,

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The product graph G m * G p of two given graphs G m and This work deals with product graphs for which we provide bounds for the connectivity parameter κ. Moreover, we state sufficient conditions that guarantee these product graphs to be maximally connected or superconnected. As a consequence, we deduce that even small networks with low reliability may lead to larger networks with high levels of fault-tolerance.

A permutation graph (or generalized prism) G π of a graph G is obtained by taking two disjoint copies of G and adding an arbitrary matching between the two copies. Permutation graphs can be seen as suitable models for building larger interconnection networks from smaller ones without increasing significantly their maximum transmission delays, in such a way that these larger networks are highly fault-tolerant. For permutations graphs, in this article we provide conditions that guarantee optimal values for two parameters of connectivity, λ and κ . For a connected graph G the restricted edge-connectivity λ (G) is defined as the minimum cardinality of a restricted edge-cut; that is, the minimum cardinality of a set S of edges such that G − S is not connected and S does not contain the set of incident edges of any vertex of the graph. A graph G is said to be λ -optimal if λ (G) = ξ (G), where ξ (G) is the minimum edge-degree in G defined as ξ (G) = min{d(u) + d(v) − 2 : uv ∈ E(G)}, and d(u) denotes the degree of vertex u. Among other things, we prove that permutation graphs satisfy:if G is triangle-free. We also study the vertex case considering the restricted connectivity κ (G) and relating it to the superconnectivity κ 1 (G); the latter is defined as the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has at least two vertices. For instance, we prove that 2κ(G) ≤ κ 1 (G) ≤ κ (G π ) ≤ ξ (G π ) if G is triangle-free and the permutation graph has no cycles of length five.