What you are describing is sometimes goes under the phrase *reflexive* vs. *irreflexive* graphs. What you seem to be describing are two categories:

**Irreflexive graphs**: the objects are sets with irreflexive symmetric relations, the morphisms are relation-preserving functions on the underlying sets.

**Reflexive graphs**: the objects are sets with reflexive symmetric relations, the morphisms are relation-preserving functions on the underlying sets.

It is fun to compute cartesian products in both of these and to discover the two well-known kinds of graph products. Graph theorists could benefit from a bit of category theory.

There are actually many variations on categories of graphs. Here are some others.

### Graphs as relations

The categories of relations and relation-preserving functions are categories of simple graphs. We can require various additional properties. Symmetric relations give us symmetric graphs, reflexive relations give us graphs in which morphisms are allowed to squish edges, etc.

### Directed graphs

This is perhaps the most straightforward example. Let $\mathcal{C}$ be the category with two objects, called $\mathsf{E}$ and $\mathsf{V}$ and two arrows $\mathsf{s}, \mathsf{t} : \mathsf{E} \to \mathsf{V}$. Then the (covariant) presheaf category $\mathbf{Set}^\mathcal{C}$ is just the category of directed graphs. An object of the category can be viewed as a pair of sets $(E, V)$ together with two maps $s, t : E \to V$.

### Reflexive directed graphs

We can play with directed graphs. For example, if we add an arrow $\mathsf{r} : \mathsf{V} \to \mathsf{E}$ which is a common right inverse of $\mathsf{s}$ and $\mathsf{t}$, we obtain directed graphs with chosen loops.

### Symmetric graphs

Another option is to add an arrow $\mathsf{o} : \mathsf{E} \to \mathsf{E}$ satisfying $\mathsf{t} \circ \mathsf{o} = \mathsf{s}$ and $\mathsf{o} \circ \mathsf{o} = \mathsf{o}$ (yes, I am using different kinds of circles on purposes to confuse you). The presheaf category then corresponds to directed graphs in which each edge $e : a \to b$ has an opposite $o(e) : b \to a$, and "opposite" is an involution. It may happen that a loop is its own opposite.

### Graphs as monoid actions

To get more fun out of categories of graphs we can do the following. Let $M$ be the monoid of functions $\lbrace 0, 1 \rbrace \to \lbrace 0, 1 \rbrace$. There are four elements, the identity $1$, the map $t$ which swaps $0$ and $1$, and the two constant maps $0$ and $1$. The category of right $M$-actions can be seen as a category of graphs.

Suppose $(X, m : X \times M \to X)$ is a right action on $X$. This is a graph in the following sense. The vertices are those elements of $X$ that are fixed by the action of $0$, or equivalently by the action of $1$. We do not have edges, but rather "half-edges". A half-edge is an element of $X$ which is not a vertex. Each half edge $e$ has its mate $e \cdot t$, and together you can think of them as a wholesome edge (unordered). A half-edge $e$ is attached to the vertex $e \cdot 0$, while its mate is attached to $e \cdot 1$.

We almost get the usual symmetric (non-simple) graphs, except that there can be degenerate half-edges that are their own mates (and therefore necessarily loop-like).

The relevant reference for this answer is Bill Lawvere's Categories of spaces may not be generalized spaces as exemplified by directed graphs.

for what, drawbacks when trying to dowhat? Definitions do not exit in the vacuum; they are worth their usefulness in gold. What are you trying to achieve? $\endgroup$ – Mariano Suárez-Álvarez Nov 12 '12 at 5:14whydo you want there to be a morphism between the graphs where there isn't? That someone somewhere has preferred the well-known definition of morphism is worth-knowing, but a motivation foryourquestion would not be bad, either! $\endgroup$ – Mariano Suárez-Álvarez Nov 12 '12 at 16:12