Minimal configuration bicyclic graphs

The nullity η(G) of a graph G is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. If η(G)?=?1, then the core of G is the subgraph induced by the vertices associated with the nonzero entries of the kernel eigenvector. The set of vertices which are not in the core is the periphery of G. A graph G with nullity one is minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. An ∞-graph ∞(p,?l,?q) is a graph obtained by joining two vertex-disjoint cycles C _p and C _q by a path of length l?≥?0. Let ?* be the class of bicyclic graphs with an ∞-graph as an induced subgraph. In this article, we characterize the graphs in ?* which are minimal configurations.