The MakeWeighted command returns a graph with vertices and edges from G. If M is part of the input, then the edge weights are taken from it; otherwise edge weights are assumed to be 1. If G is undirected, then M is assumed to be a symmetric matrix.

For efficiency, use datatype=integer for wordsize integer weights and datatype=float[8] for numerical (decimal) edge weights.

To read or modify the edge weights of a weighted graph, use the GetEdgeWeight and SetEdgeWeight commands.

with(GraphTheory):

G := Graph({{1,2},{2,3},{3,1}});

$G≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 3\; vertices\; and\; 3\; edge(s)}$

M := Matrix([[0,2,3],[2,0,1],[3,1,0]]);

$M≔\left[\begin{array}{ccc}0& 2& 3\\ 2& 0& 1\\ 3& 1& 0\end{array}\right]$

H1 := MakeWeighted(G, M);

$\mathrm{H1}≔\mathrm{Graph\; 2:\; an\; undirected\; weighted\; graph\; with\; 3\; vertices\; and\; 3\; edge(s)}$

Edges(H1);

$\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{2\,3\right\}\right\}$

WeightMatrix(H1);

$\left[\begin{array}{ccc}0& 2& 3\\ 2& 0& 1\\ 3& 1& 0\end{array}\right]$

M := Matrix(M,datatype=float[8],shape=symmetric);

$M≔\left[\begin{array}{ccc}0.& 2.& 3.\\ 2.& 0.& 1.\\ 3.& 1.& 0.\end{array}\right]$

H2 := MakeWeighted(G,M);

$\mathrm{H2}≔\mathrm{Graph\; 3:\; an\; undirected\; weighted\; graph\; with\; 3\; vertices\; and\; 3\; edge(s)}$

WeightMatrix(H2);

$\left[\begin{array}{ccc}0.& 2.& 3.\\ 2.& 0.& 1.\\ 3.& 1.& 0.\end{array}\right]$