**Lemma 2.** *If there is a node within the competition radius area of another node, then the presence of two CHs is impossible.*

**Proof of Lemma 2.** According to Equation (7), each node has a unique delay time. If a node has a lesser delay time, it transmits head\_msg with *Rc*(*i*) radius and all other nodes with this radius change their own position to CM. Therefore, if there are two nodes located within competition radius, it is not possible to have more than two CHs. -

**Lemma 3.** *After performing HCD, whole network space is covered by CHs; in other words, a node (either CH or CM) will be in the same cluster.*

**Proof of Lemma 3.** If a node such as A is neither a CH nor a CM after performing HCD, node A will have two positions before executing the algorithm: (1) it does not have any neighbors and (2) has a neighbor. In the first case, according to the algorithm, node A announces to be the cluster head at the end of the delay time. In the second position, given that node A is a CH nor a CM, none of the neighbor nodes broadcast the CH message. If one of them was CH, then A will be a CM. However, if A and some of its neighbors are not CH and they do not join to a cluster, then one of them will be converted to the CH after the end of delay time. Subsequently, a message is broadcasted and neighbor nodes receiving the message will change their positions to the CMs. Therefore, node A will be either a CH or will join another CH before the end of the algorithm. Hence, it can be said that the network is completely covered by CHs. -

**Lemma 4.** *The HCD guarantees the load balancing of CHs in intracluster and intercluster multihop routing.*

**Proof of Lemma 4.** According to the Equation (4), it can be demonstrated that the node radius depends on its distance to the BS and its remaining energy. Therefore, CHs having more distance to BS and higher remaining energy make larger clusters and they have more CMs. Hence, it consumes more energy to receive and collect data from CMs. Moreover, ACH nodes are applied to maximize the stability and balanced energy consumption of CMs during data transmission to the CH and also to balance CH energy consumption for receiving the data. On the other hand, CHs close to the BS with less energy have a smaller radius and less CMs. Therefore, they save more energy for receiving and collecting data from the CMs, and they guarantee the load balancing of the HCD protocol. -
