Weight Function for a Graph
A weight function for a graph assigns a weight or cost value to each edge. It represents numeric properties of the connections between vertices.
Some examples of graph weight functions:
 Shortest path weights  Travel time or distance
 Betweenness centrality  Number of shortest paths through edge
 Probabilistic weights  Likelihood of traversing edge
Java example  Random weight assignment:


C++ example  Distance weights:


Python example  Betweenness centrality:


Weight functions allow modeling edge costs and numeric properties in graphs. Useful for optimization problems.
In graph theory, a weight function assigns a value, often referred to as “weight,” to each edge in the graph. This weight can represent distances, costs, or any other measure that one might want to optimize. For example, in a road network, the weight could represent the distance or time required to traverse from one city to another. Weighted graphs are often used in algorithms like Dijkstra’s shortest path, Kruskal’s algorithm, etc.
Let’s visualize a weighted graph to clarify the concept of a weight function for a graph. In a weighted graph, each edge has an assigned weight, often representing distances, costs, or other metrics.
Consider a simple graph with 4 vertices: A, B, C, and D. These vertices are connected by the following edges:
 Edge
AB
with weight5
 Edge
AC
with weight3
 Edge
BC
with weight2
 Edge
BD
with weight6
 Edge
CD
with weight4
Here’s a textual representation of this weighted graph:
A 5 B 6 D
 / 
 / 
3 2 4
 / 
C 
The weight function ( w: E \rightarrow \mathbb{R} ) maps each edge to a real number, representing its weight:
 ( w(AB) = 5 )
 ( w(AC) = 3 )
 ( w(BC) = 2 )
 ( w(BD) = 6 )
 ( w(CD) = 4 )
This weight function is often a key part of algorithms that solve problems like finding the shortest path or minimum spanning tree.
The concept of a weight function in a graph assigns a numerical value to each edge in the graph. These values are often called “weights.”
Visual Representation:
Let’s consider a simple weighted graph with vertices A, B, C, and D and weights on the edges as follows:
A(2)B
 
(1) (3)
 
D(4)C
In this graph:
 The edge between A and B has a weight of 2.
 The edge between B and C has a weight of 3.
 The edge between C and D has a weight of 4.
 The edge between D and A has a weight of 1.
The weight function w
can be represented as a set of tuples:
 w(A, B) = 2
 w(B, C) = 3
 w(C, D) = 4
 w(D, A) = 1
Key Takeaway:
A weight function assigns numerical values to the edges of a graph. It’s commonly used in problems involving shortest paths, network flow, and many other optimization problems in graph theory.
Solution
Below are code examples that demonstrate how to implement a weight function for a graph in Java, C++, and Python.
Java


C++


Python


Key Takeaways
 A weight function assigns a value to each edge in the graph.
 Weighted graphs are foundational in several important graph algorithms.
 The code examples use a nested map (Java, C++) or dictionary (Python) to hold the weight values for each edge.