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This paper explores the stability analysis problem for nonlinear systems which have general linear feedback interconnections. Systems are often modeled in this manner in the study of decentralized control because many communication topologies can be modeled and analyzed using connections to graph theory. We present necessary conditions for stability of a classification of interconnected systems, and we give some examples to provide insight into this problem. These conditions are related to positive definiteness of matrices associated with the feedback interconnection, and specialize to the common case where the Laplacian matrix of a graph represents the communication topology of the system.