Dijkstra single-source shortest path $\Omega(n\log n)$?
Let $m$ be the number of vertices and $n$ the number of edges. The best you can get is $O(n + m \log m)$ by using a Fibonacci heap.

A Fibonacci heap allows decreasing the key of an element in amortized constant time. For each edge in the graph, Dijkstra's algorithm executes at most one decrease key operation. All these operations take time $O(n)$ in total. On top of that we have $m$ minimum extractions, which take time $O(m \log m)$ in total. Therefore the final complexity is $O(n + m \log m)$.