Existence of edge in graph

Question

I want to know whether an edge exist between two specific vertex? Is there any function like EdgeExist[g,{v1,v2}]?

Answer 1

Yes, there is EdgeQ:

g = Graph[{1 -> 2, 2 -> 3, 3 -> 1}, VertexShapeFunction -> "Name"]

{EdgeQ[g, 1 -> 2], EdgeQ[g, DirectedEdge[1, 2]], 
 EdgeQ[g, 1 <-> 2], EdgeQ[g, UndirectedEdge[1, 2]],
 EdgeQ[g, 2 -> 1], EdgeQ[g, DirectedEdge[2, 1]]}

{True, True, False, False, False, False}

If you need a function that takes a list of (unordered) vertices as the second arguments, you can use

edgeQ[g_, e : {_, _}] := Or @@ (EdgeQ[g, #] & /@ {DirectedEdge @@ e, 
     DirectedEdge @@ Reverse[e], UndirectedEdge @@ e})

edgeQ[g, {1, 2}]

True

If the ordering is important, remove DirectedEdge @@ Reverse[e] from the function definition.

Notes: As @Szabolcs noted in the comments, a <-> b is equivalent to UndirectedEdge, but a -> b is not the same as DirectedEdge. a -> b is treated as DirectedEdge only in a directed graph; in an undirected or mixed graph it is treated as UndirectedEdge.