Dear Peter,

I made it in a better way :)

But, I have not received your final answer of my previous question yet.

It is still open.

Dear Peter,

This sentence has not made by me.

I have received this sentence from one other scientist.

That's why I also do not understand the meaning :)

Is there anyone here who can understand the meaning of the mentioned sentence?

Thank you for solving my problem as well :)

Ok, I think the answer is this:

A graph peoperty is monoton if every subgraph of a graph with property P, also has the property P. Same as bipartite graphs. every subgraph (Withot isolated vertices), of a bipartite graph is also bipartite.

Now, the meaning of "Monotonically should make no difference" is,  for each of the graphs G in the class, if H is a subgraph of G, then H should have the same properties as G.

@Nazanin "A graph peoperty is monoton if every subgraph of a graph with property P, also has the property P"  is called "hereditary property".  

As for the original (part of a) sentence: if we join it to another domain of mathematics, it makes sense such as it is, despite imprecise wording: e.g. in a class of exponential functions with a basis a>1... 

@Viera, thank you for your explanation. I think monotone and hereditary are both same. Please look at here:

https://en.wikipedia.org/wiki/Graph_property 

The "Properties of properties" part.

@Peter, thanks for your helps. It is not in Italic. So we should think more about the meaning. But I think my explanation is 70% correct.

Please send me your ideas in case of any new explanation.

@Peter Breuer . very well answered.