Radio Number of the Cartesian Product of Stars and Middle Graph of Cycles

With the rapid development of wireless communication network, channel is the most important part in the communication process, and the problem of reasonable channel assignment becomes increasingly serious. To solve this problem, we study the radio label problem of graph. The graph is usually used as the channel assignment modeling of wireless communication, and the channel assignment problem of the network is simulated by the vertex labeling problem of the graph. Various applications of radio labeling, such as frequency assignment in mobile communication systems, signal processing, Parallel and distributed computing, circuit and sensor network design, play an important role in the channel assignment process of wireless communication networks. The channel assignment in the network is converted to the vertex labeling problem of the graph. The maximum radio label of the graph is called its span, and the minimum possible span is called the radio number of the graph. The aim is to find an optimal radio label to reduce the channel utilization rate in the network, so as to reduce the interference in the process of network communication. In this paper, we mainly study the topology of the Cartesian product of stars with $\mathbf{n}$ vertices and the middle graph of cycles, where $m \geq 3$. We simulate the channel assignment of a wireless communication network with the same structure as it, obtain the lower bound of its radio label, and determine optimal radio label.