Reliability analysis of Cayley graphs generated by transpositions.

Let Γn be the symmetric group on {1,2,…,n} and S be the generating set of Γn. The corresponding Cayley graph is denoted by Γn(S). If all elements of S are transpositions, a simple way to depict S is via a graph, called the transposition generating graph of S, denoted by A(S) (or say simply A), where the vertex set of A is {1,2,…,n}, there is an edge in A between i and j if and only if the transposition (ij)∈S, and Γn(S) is called a Cayley graph obtained from a transposition generating graph A. In this paper, by exploring and utilizing the structural properties of these Cayley graphs, we obtain that the pessimistic diagnosability of Γn(S) is equal to 2|E(A)|−2 if A has no triangles or 2|E(A)|−3 if A has a triangle. As corollaries, the pessimistic diagnosability of many kinds of graphs such as Cayley graphs generated by unicyclic graphs, wheel graphs, complete graphs, and tree graphs is obtained.