The adjacency matrix of a network that has $$N$$ nodes has $$N$$ rows and $$N$$ columns. If there is a link from node $$i$$ to node $$j$$, then $$A_ij = 1$$. If node $$i$$ and node $$j$$ are not connected, then $$A_ij = 0$$.

For an undirected network, the link $$(i, j)$$ has two representations: $$A_ij$$ = $$A_ji$$. That’s why the adjacency matrix of an undirected network is always symmetric.

Lets have a look at the adjacency matrix of g1 and g2.

For g1:

par(mar = c(1, 1, 1, 1))
plot(g1) as_adjacency_matrix(g1)
## 4 x 4 sparse Matrix of class "dgCMatrix"
##
## [1,] . 1 1 .
## [2,] 1 . 1 .
## [3,] 1 1 . 1
## [4,] . . 1 .

For g2:

par(mar = c(1, 1, 1, 1))
plot(g2) as_adjacency_matrix(g2)
## 4 x 4 sparse Matrix of class "dgCMatrix"
##
## [1,] . 1 1 .
## [2,] . . 1 .
## [3,] . . . 1
## [4,] . . . .

Each dot in the matrix means $$0$$. The adjacency matrices here did not label the column name. To better understand it, you can imagine that the columns are also labeled as [1, ], [2, ], [3, ] and [4, ] as the rows are.