Empty and trivial graph [Diestel's book] 
The number of vertices of a graph $G$ is its order, written as $|G|$;
For the empty graph $(\varnothing, \varnothing)$ we simply write
$\varnothing$. A graph of order $0$ or $1$ is called trivial.


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*As far as I understand the graph is called empty if both vertex set and edge set are empty, right?


*What is difference between graph of order $0$ and empty graph?


*What does mean that graph is non-empty?
Sorry if I am asking dumb questions but I'd like to understand the difference between them.
 A: A graph of order $0$ has no vertices and (necessarily) no edges. It's an empty graph.
A graph of order $1$ has one vertex and (necessarily) no edges, since an edge must join two distinct vertices (but sometimes graphs are allowed to have loops -- edges that join a vertex to itself. Whether that is OK depends on the context.)
In some contexts an empty graph means one with no vertices and (necessarily) no edges. In that case the only empty graph is the graph of order $0$. In other contexts a graph is empty just when it has no edges - it may have vertices. I don't know the context for your text.
There is no universally accepted vocabulary for graphs. Sometimes loops are allowed. Sometimes there can be multiple edges between vertices. Sometimes edges have directions. Whenever you read a graph theory paper the author will explain how they use these terms.
A: A non empty graph is a pair $(V,E)$ where $V$ is a non empty set (whose elements are known as vertices) and $E$ is a subset of the set of all two element subsets of $V$ (whose elements are known as edges) .  The order  of a graph is simply the number(cardinality in the infinite case) of vertices it has . If $V$ is the empty set then so is $E$ and the graph is called an empty graph. A graph of order $0$ is empty this immediately follows from the definition.
