Bellman-Ford Equation

Let there b a graph where $(x,y)$ is the edge from vertex $x$ to vertex $y$ and $d_x(y)$ is the least-cost path from node $x$ to $y$.

Finally let $v$ be the neighbour of $x$, i.e. the node connected by a single edge to $x$.

Then the least costs are related by $$\begin{gathered} d_x(y)={\min }_v \\ c(x,v),d_v(y) \end{gathered}$$ which is the Bellman-Ford equation.