I am struggling to understand the concept of isomorphism. By definition, if G and H are two simple graphs so that V(g) and V(h) are the number of nodes in G and H respectively, then isomorphism is defined as a function from f:V(G)->V(H) with the following properties: (i) f being bot surjective and injective (thus, f is a bijection) (ii) the nodes u and v are neighbours/adjacent in G iff the nodes f(u) and f(v) are adjacent in H.
Now, for the problem.
Draw a complete graph with five nodes. For this, I understand that a complete graph has each vertex adjacent to all other vertices in the graph. (See the illustration).
Then, is it possible to find another (non-isomorphic!) complete graph with five nodes? How do I approach this question?
Intuitively, I think that it is not possible, since a complete graph implies that every pair of distinct vertices ought to be connected by a unique edge, and thus, all possibilities are exhausted. However, this is not based on any written work. How can I approach the question?