Reliabilities for two kinds of graphs with smaller diameters.

• Owing to the great difference of different networks connection ways, most of the research on their performances is carried out separately by using special methods. We propose two kinds of recursive networks based on connected graphs such that they contain diverse networks, aimed to study their performances by a more unified method. • We prove that the two kinds of recursive networks have common superior performances: smaller diameters and higher reliabilities—maximal connectivity and diagnosability. • We show that our results can be extended to a number of networks. Two crucial parameters to be considered when evaluating a network's reliability are its connectivity and diagnosability. The connectivity of a graph is defined as the smallest number of vertices that, if removed, would leave a disconnected or single-vertex graph. The diagnosability is the maximum number of faulty vertices that the system can correctly identify. In this paper, aim to obtain the reliability of various graphs (e.g. interconnection networks and data center networks), we propose two kinds of graphs (resp. networks)—matching recursive graphs (MRGs) and recursive networks based on connected graphs (RNCGs). We prove that these two kinds of graphs have smaller diameters. Furthermore, we study their reliabilities, including connectivities and diagnosabilities. Moreover, we also extend our results to many graphs. [ABSTRACT FROM AUTHOR]