Adjacency matrices of multigraphs 

Should the entry (adjacency matrix) for row = g, column = g be "2" instead of 1?
Also, should the entry (incidence matrix) for row = g, column = e11 be "2" instead of 1?
I have one lecturer saying that both entries should be "1" and another lecturer saying these two entries should have "2". This looks like it should be obvious; but, I've been given conflicting information about these entries and want to check if there is a "right" answer.
 A: Consider the degree sum formula for a graph $G = (V,E)$,
$$\sum_{v \in V} \text{degree} \ v = 2|E|.$$
Now take a look at the multi-graph $G = (\{1\},\{1,1\})$. We obviously get a contradiction if we work with this definition of a self-loop ($\text{degree} \ v = 1$ for edge $\{v,v\}$).
A: If there is a loop then the entry should be 2 instead of 1. For the incidence matrix, the sum of each column must always be 2.
A: Recall the definition of an adjacency and incidence matrix.
In an adjacency matrix $A$, the rows and columns are indicated by vertices, and the entry $a_{ij}$ is the number of edges from vertices $v_i$ to $v_j$.
In an incidence matrix $B$, the rows are indicated by the vertices of the graph, and the columns are indicated by the edges of a graph, and the entry $b_{ij}$ is the number of times vertex $i$ is incident to edge $j$. In a directed graph, the convention of the outvertex contributing a +1 and an invertex contributing a -1 is usually adopted.
I presume you are asking what are the entries for a loop in these matrices.
We first approach the adjacency matrix. 
The entry $A_{ii}$ in the adjacency matrix will be 2 in an undirected graph, viewing the start and end points as 2 different objects, rather than the same vertex. This is necessary for the degree-sum formula to be satisfied. 
In directed graphs (unless both directions are indicated), this entry will be 1. Similarly, the degree-sum for directed graphs is also satisfied here.
For the incidence matrix, the following convention is usually adopted. For a loop $j$ on vertex $i$, the entry $B_{ij}$ will be 2 in an undirected graph and 0 in a directed one. This is similar in the previous case where we treat the start and end points as two different objects.
