How do I set properties of graphs permanently? Is there any way?
For example VertexLabels -> "Name".
I would like to see labels, always when I open a new notebook and I run the codes: CycleGraph[5], WheelGraph[6], MyCommand[...], Graph[...], etc. I.e.: Object with Head=Graph or a function which return such an object should display labels.

SetOptions[Graph, VertexLabels -> {3 -> "Name"}]followed by anyGraph: you'll get the same label every time, unless you override it. – Jens May 07 '16 at 22:02CycleGraph[5]. – ZYX May 07 '16 at 22:19