trjhtr

I'm using ArrayList<Nodes> to store adjacent nodes, and ArrayList<Integer> to store the corresponding weights of the vertices. Is this efficient if I only use it for small inputs? Is it efficient for large inputs as well?

class Node 
 public ArrayList<Node> adjacent = new ArrayList<Node>();
 public ArrayList<Integer> weights = new ArrayList<Integer>();
 boolean visited;

Is this efficient if I only use it for small inputs?

For small inputs, anything is efficient.
For big data, having each Node as such a large object would be inefficient.
I depends what kind of actions you would do over the graph. If you access it sequentially, there can be also a linked implementation like:

class LinkedNode 
 public Node firstChild; 
 public Node nextSibling; 
 boolean visited;

Here's an alternative, without objects but with arrays instead:

int edgeWeights;
int edgeIndicies;

At the begin, edgeIndicies will have values [0,1,2,...n].
Sorting the weights, you move the indicies instead, so if the indicies are the following:
[5,7,9,...] you know edgeWeights[5] is the smallest, edgeWeights[7] is the second smallest, ...etc.