The difference between dense graph and sparse one How to decide whether the following statement is correct? 

O(E log E) and O(E log V) are equivalent regardless whether graph is dense or
sparse
You have to be careful with your definitions. For any graph, the number of nodes and the number of edges is a constant, so $O(V) = O(E) = (1)$. This is clearly not helpful.

Sparse and dense only make sense for a family of graphs $\{G_n\}_{n=1}^\infty$.

The edge density of a graph $G$ is the proportion of possible edges that a graph actually has:

$$ D = \frac{|E(G)|}{{|V(G)|}\choose{2}}. $$

The edge density of a family of graphs is

$$ D = \lim_{n \to \infty} \frac{|E(G_n)|}{{|V(G_n)|}\choose{2}}. $$

The family is sparse if $D = 0$ and dense if $D &gt; 0$. You must analyze the performance of your algorithm under either type of family as inputs.

This definition is from Diestel's Graph Theory.