We consider the reliability of regular graphs of degree k in which the vertices of the graph fail independently with probability p. The probability of disconnection is minimized for small values of p if the graph has connectivity k and has the smallest number of vertex cut sets with k vertices. In this paper, we show how to construct such graphs.

A graph G is called invertible if its adjacency matrix A has an inverse which is the adjacency matrix of some graph H. All such graphs were shown by Haraq and Minc to have the form nKz. We now introduce signed invertible (or briefly s-invertible) graphs G as those whose inverse H is a signed graph. We identify two infinite classes of s-invertible graphs: the paths PZn of even order, and the corona of any graph with KI. We then characterize s-invertible trees. INVERTIBLE GRAPHSLet A = A(G) be the adjacency matrix of a graph G . The matrix A may or may not have an inverse. When it does, we say that graph G is invertible if A-' is the adjacency matrix of some graph H . In this case, H is called the inverse graph of G . Obviously, whether G is invertible is independent of the particular labeling of the vertices used to obtain A.If A(G) has an inverse which is the adjacency matrix of a signed graph H , we say that G is signed invertible, or briefly s-invertible, and H is the signed inverse of C .Clearly, H is also independent of the labeling used for G. Our purpose is to study invertible and s-invertible graphs. We follow the terminology and notation of the book Obviously, an integral matrix M has an integral inverse if and only if det M = 2 1.Therefore,the graphs we are considering must satisfy det A = k 1. If two nonadjacent vertices in a graph have the same neighborhood, then det A = 0 as two rows of A would be identical. Thus, a complete bipartite graph Km," has zero [31.

If a practical network is modelled as a graph in which the lines are perfect but the points may fail then a primary measure of vulnerability is the point connectivity of the graph and one possible secondary measure of vulnerability is the number of minimum point disconnecting sets. The graph should have the maximum possible point connectivity and the minimum number of point disconnecting sets. A lower bound for the number of minimum point disconnecting sets is derived by identifying points with identical adjacencies.

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