This section is about isomorphisms of graphs, whereby the term "isomorphism" is used in both of its meanings (according to the Meriam-Webster dictionary, see https://www.merriam-webster.com/dictionary/isomorphism): "1: the quality or state of being isomorphic." and "2: a one-to-one correspondence between two mathematical sets".
At first, an operation is defined (see df-grim) which provides the graph isomorphisms (as "one-to-one correspondence") between two given graphs. This definition, however, is applicable for any two sets, but is meaningful only if these sets have "vertices" and "edges".
Afterwards, a binary relation is defined (see df-gric) which is true for two graphs iff there is a graph isomorphisms between these graphs. Then these graphs are called "isomorphic". Therefore, this relation is also called "is isomorphic to" relation. More formally, resp. . Notice that there can be multiple isomorphisms between two graphs. For example, let and be two graphs with two vertices and one edge, then and are two different isomorphisms between these graphs.
The names and symbols are chosen analogously to group isomorphisms (see df-gim) resp. isomorphism between groups (see df-gic).
The general definition of graph isomorphisms and the relation "is isomorphic to" for graphs is specialized for simple hypergraphs (gricushgr) and simple pseudographs (gricuspgr). The latter corresponds to the definition in [Bollobas] p. 3. It is shown that the relation "is isomorphic to" for graphs is an equivalence relation, see gricer. Finally, isomorphic graphs with different representations are studied (opstrgric, ushggricedg).
Another approach could be to define a category of graphs (there are maybe multiple ones), where graph morphisms are couples consisting of a function on vertices and a function on edges with required compatibilities, as used in the definition of . And then, a graph isomorphism is defined as an isomorphism in the category of graphs (something like "GraphIsom = ( Iso ` GraphCat )" ). Then general category theory theorems could be used, e.g., to show that graph isomorphism is an equivalence relation.