To secure IoT and medical sensor networks, mechanisms that are able to detect malicious and compromised devices or unreliable devices are needed. In this study a graph-theoretic trust model is suggested, where the sensors are the nodes and the communication relationship between the sensors is represented by a weighted edge. The trust is estimated directly from packet delivery, data consistency, responsiveness and authentication behavior, and indirectly is based on a collection of the recommendations of neighboring nodes. A composite score adaptively weights both direct, indirect and historical evidence and contextual factors. The operations performed on the nodes (adjacency-matrix operations) and the analysis of paths (path-based analysis) both enable scalable propagation of trust and isolation of suspicious nodes. The simulation results show better detection of malicious nodes, reliability of packets and the robustness of the network when compared to the traditional reputation methods. The framework offers an easy-to-understand and lightweight basis for secure distributed healthcare monitoring applications.