# Graphs in space efficient representation

Let $$G$$ be a graph such that $$V$$ denotes a vertex set and $$E$$ is an edge set of the graph $$G$$. Let us consider that for the input graph $$G$$ it is the case that $$|E| \le O(|V| \log |V|)$$. Given a graph $$G$$, I want to answer whether $$(u,v) \in E$$. If the input graph is given by adjacency representation then I can answer whether $$(u,v) \in E$$ in $$O(\log |V|)$$ time with space complexity $$O(|V| \log |V|)$$.

Space complexity refers to the space taken by input in words. One word means $$O(\log |V|)$$ bits.

Question: Does there exists a representation of graph $$G$$ such that I can answer whether $$(u,v) \in E$$ in $$o(\log |V|)$$ time with space complexity $$O(|V|)$$?