A graph property $Q$ is called monotone increasing if $\forall$ graph $G \in Q$, we have $G + e \in Q$, where $e \in K_n $\ $ G$ , ($K_n $ being Complete graph). So basically it is preserved by the addition of edges.
Example - Connectivity.
Likewise the term monotone decreasing can be defined as follow - A graph property $Q'$ is called monotone decreasing if if $\forall$ graph $G \in Q$, we have $G - e \in Q$.
Example - Graphs with at least one isolated vertex.
But I am not able to figure out any graph property which is neither increasing nor decreasing. Any help ?
How about regularity? Being 3-regular (where every vertex has 3 edges) is neither monotone increasing or monotone decreasing. Other examples might include being bipartite, or being a tree (with the understanding that a tree is a connected acyclic graph), or other shape-dependent properties.