Implementing Dijkstra's algorithm in Neo4j

Yes it is possible with Cypher or with a dedicated endpoint of the Neo4j ReST API.

BTW, the examples from the Cypher Neo4j documentation are self explaining :

http://neo4j.com/docs/milestone/query-match.html#_shortest_path

To get the shortestPath between two nodes :

MATCH (from: Location {LocationName:"x"}), (to: Location {LocationName:"y"}) , 
path = shortestPath((from)-[:CONNECTED_TO*]->(to))
RETURN path

If you want to get all the shortest

MATCH (from: Location {LocationName:"x"}), (to: Location {LocationName:"y"}) , 
paths = allShortestPaths((from)-[:CONNECTED_TO*]->(to))
RETURN paths

If you want to order by the length (number of hops) of the paths in descending order :

MATCH (from: Location {LocationName:"x"}), (to: Location {LocationName:"y"}) , 
paths = allShortestPaths((from)-[:CONNECTED_TO*]->(to))
RETURN paths
ORDER BY length(paths) DESC

If you to to get the shortestPath + the sum of the relationship distance property :

MATCH (from: Location {LocationName:"x"}), (to: Location {LocationName:"y"}) , 
path = shortestPath((from)-[:CONNECTED_TO*]->(to))
WITH REDUCE(dist = 0, rel in rels(path) | dist + rel.distance) AS distance, p
RETURN p, distance