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Going through Graph theory , i found

We call Graph $G$ edge Maximal if with a given graph property if $G$ itself has the property but no graph $G$+$xy$ does for non adjacent vertices $x,y$ $\epsilon$ G

I am not getting what does it really mean .!

please help me out !!!

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    It means that $G$ itself has the property, but if you add a new edge to $G$ then it will no longer have the property. – 2017-01-27

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A graph with a certain property is called edge maximal for that property if you cannot add another edge but keep the property. For instance, a tree is an edge-maximal cycle-free graph. You cannot add an edge while keeping it cycle-free, because adding an edge to a tree always adds a cycle. Similarly, if you graph consists of two components, each of which is a complete graph, then this graph is edge maximal disconnected: adding any edge to the graph turns it into a connected graph.

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    This is definitely a concept best explained using examples in my opinion. – 2017-01-27
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    Yes, even better explained using pictures. – 2017-06-22