weighted graph

a weighted graph is $\text{[math]}$ such that $\text{[math]}$ is a graph (directed or not) and $\text{[math]}$ is a weight function that gives each edge $\text{[math]}$ a value $\text{[math]}$ . this value is called the weight or length of the edge $\text{[math]}$

path length:
let $\text{[math]}$ be a weighted graph, let $\text{[math]}$ be a path in $\text{[math]}$ , the length of the path $\text{[math]}$ is defined as the sum of weights of the edges in the path:
$\text{[math]}$

distance between vertices:
for all $\text{[math]}$ , the distance from $\text{[math]}$ to $\text{[math]}$ is defined as the length of the minimal path from $\text{[math]}$ to $\text{[math]}$ , i.e. $\text{[math]}$ . (same definition as for unweighted graphs)