In this talk, I will introduce some research topics in interconnection networks that are interesting both for mathematicians and computer scientists.  Any interconnection network can be viewed as a graph.  We are particularly interested in finding those graphs that have nice properties in the design of interconnection networks.  Some of the properties of the graphs are diameters, degrees, connectivities, and fault tolerance.  I will describe some of my recent works on this subject and propose some open problems which I believe that are interesting.