Connected healthcare is the process of integrating healthcare smart applications into smart devices. These systems can enable better patient-hospital experience, efficient time usage, reduced errors, safety and security, and ultimately improved treatments. These smart devices which form an IoT network are extremely dynamic because of the user movement. In an environment where there is a constant change in the network topology and its traffic profile, it is a challenging task to provide reliable network connectivity and to maintain the IoT network. Therefore, ensuring healthcare traffic is resilient towards change in the traffic profile is of paramount importance. This paper leverages graph theory concepts to understand the behaviour of the healthcare IoT network. The paper highlights the importance of the PN (PN) and traffic splitting (stratification). A PN is a node in the network which has enough computing resources to share with other network devices. By optimizing the selection of PN, the drastic improvement in the network performance could be achieved. Moreover, we show that splitting the traffic along with optimized PN selection minimizes the chances of healthcare traffic drop during the period of high network usage.