On average fault-tolerance in product graphs

Let G be a graph with vertex set V = V (G) and edge set E = E(G). The cardinalities of these sets are denoted by |V (G)| = n and |E(G)| = e. Let u and v be two distinct vertices of G. A path from u to v, also called an uv-path in G, is a subgraph P with vertex set V (P ) = {u = x0, x1, . . . , xr = v} and it is usually denoted by P : x0x1 · · ·xr. Two uv-paths P and Q are said to be internally disjoint if V (P ) ∩ V (Q) = {u, v}. A cycle in G is a path C : x0x1 · · ·xr such that x0 = xr. The girth of G, denoted by g(G), is the length of a shortest cycle in G, and if G contains no cycles, then g(G) = ∞. The set of adjacent vertices to v ∈ V (G) is denoted by NG(v). The degree of v is dG(v) = |NG(v)|, whereas δ(G) = minv∈V (G) dG(v) and d(G) = 1 n ∑