Graph operations produce new graphs from initial ones. They may be separated into the following major categories.
Unary operations create a new graph from one initial one.
Elementary operations or editing operations create a new graph from one initial one by a simple local change, such as addition or deletion of a vertex or of an edge, merging and splitting of vertices, edge contraction, etc. The graph edit distance between a pair of graphs is the minimum number of elementary operations required to transform one graph into the other.
Advanced operations create a new graph from one initial one by a complex changes, such as:
- transpose graph;
- complement graph;
- line graph;
- graph minor;
- graph rewriting;
- power of graph;
- dual graph;
- medial graph;
- Y-Δ transform;
Binary operations create a new graph from two initial ones G1 = (V1, E1) and G2 = (V2, E2), such as:
- graph union: G1 ∪ G2 = (V1 ∪ V2, E1 ∪ E2). When V1 and V2 are disjoint, the graph union is referred to as the disjoint graph union, and denoted G1 ⊕ G2;
- graph intersection: G1 ∩ G2 = (V1 ∩ V2, E1 ∩ E2);
- graph join: graph with all the edges that connect the vertices of the first graph with the vertices of the second graph. It is a commutative operation (for unlabelled graphs);
- graph products based on the cartesian product of the vertex sets:
- cartesian graph product: it is a commutative and associative operation (for unlabelled graphs),
- lexicographic graph product (or graph composition): it is an associative (for unlabelled graphs) and non-commutative operation,
- strong graph product: it is a commutative and associative operation (for unlabelled graphs),
- tensor graph product (or direct graph product, categorical graph product, cardinal graph product, Kronecker graph product): it is a commutative and associative operation (for unlabelled graphs),
- zig-zag graph product;
- graph product based on other products:
- series-parallel graph composition:
- parallel graph composition: it is a commutative operation (for unlabelled graphs),
- series graph composition: it is a non-commutative operation,
- source graph composition: it is a commutative operation (for unlabelled graphs);
- Hajós construction.
- Bondy, J. A.; Murty, U. S. R. (2008). Graph Theory. Graduate Texts in Mathematics. Springer. p. 29. ISBN 978-1-84628-969-9.
- Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.
- Reingold, O.; Vadhan, S.; Wigderson, A. (2002). "Entropy waves, the zig-zag graph product, and new constant-degree expanders". Annals of Mathematics. 155 (1): 157–187. doi:10.2307/3062153. JSTOR 3062153. MR 1888797.
- Robert Frucht and Frank Harary. "On the corona of two graphs", Aequationes Math., 4:322–324, 1970.