The AddEdge command adds one or more edges to an undirected graph. By default, the original graph is changed to a graph containing the specified set of edge(s).  By setting inplace=false the original graph remains unchanged and a new graph containing the specified set of edges is created.

If the graph is weighted, then a weighted edge can be added by calling AddEdge with one or more edges in the form $\left[\mathrm{edge}\,\mathrm{weight}\right]$, where the edge is just the set of two vertices, and the weight represents the value of the edge weight.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{CycleGraph}\left(5\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$\mathrm{AddEdge}\left(G\,\left\{1\,3\right\}\,\mathrm{inplace}=\mathrm{false}\right)$

$\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 6\; edge(s)}$

$G$

$\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$\mathrm{AddEdge}\left(G\,\left\{\left\{1\,3\right\}\,\left\{2\,4\right\}\right\}\,\mathrm{inplace}=\mathrm{false}\right)$

$\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 7\; edge(s)}$

$\mathrm{AddEdge}\left(G\,\left\{\left\{1\,3\right\}\,\left\{2\,4\right\}\right\}\right)$

$\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 7\; edge(s)}$

$G$

$\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 7\; edge(s)}$

$\mathrm{Gw}≔\mathrm{Graph}\left(\mathrm{Matrix}\left(\left[\left[0\,1\,1\,0\right]\,\left[1\,0\,0\,1\right]\,\left[1\,0\,0\,0\right]\,\left[0\,1\,0\,0\right]\right]\right)\,'\mathrm{weighted}'\right)$

$\mathrm{Gw}≔\mathrm{Graph\; 4:\; an\; undirected\; weighted\; graph\; with\; 4\; vertices\; and\; 3\; edge(s)}$

$\mathrm{Edges}\left(\mathrm{Gw}\,\mathrm{weights}\right)$

$\left\{\left[\left\{1\,2\right\}\,1\right]\,\left[\left\{1\,3\right\}\,1\right]\,\left[\left\{2\,4\right\}\,1\right]\right\}$

$\mathrm{AddEdge}\left(\mathrm{Gw}\,\left[\left\{1\,4\right\}\,2\right]\right)$

$\mathrm{Graph\; 4:\; an\; undirected\; weighted\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

$\mathrm{Edges}\left(\mathrm{Gw}\,\mathrm{weights}\right)$

$\left\{\left[\left\{1\,2\right\}\,1\right]\,\left[\left\{1\,3\right\}\,1\right]\,\left[\left\{1\,4\right\}\,2\right]\,\left[\left\{2\,4\right\}\,1\right]\right\}$

$G≔\mathrm{Graph}\left(\left[a\,b\,c\,d\right]\right)$

$G≔\mathrm{Graph\; 5:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 0\; edge(s)}$

$\mathrm{AddEdge}\left(G\,\mathrm{Trail}\left(a\,b\,c\,d\,a\right)\right)$

$\mathrm{Graph\; 5:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

$\mathrm{Edges}\left(G\right)$

$\left\{\left\{a\,b\right\}\,\left\{a\,d\right\}\,\left\{b\,c\right\}\,\left\{c\,d\right\}\right\}$