To determine whether two graphs are non-isomorphic quickly, you can use the following steps and considerations:

1. Degree Sequence Comparison: If two graphs have different degree sequences (the list of vertex degrees), they are immediately non-isomorphic. The degree sequence is a quick check because an isomorphism would imply a matching degree for each corresponding vertex in both graphs.

2. Edge Count Check: If two graphs have a different number of edges, they cannot be isomorphic. Graphs with different edge counts are structurally different, ruling out isomorphism.

3. Role of Symmetry Functions: Symmetry functions, or automorphisms, help identify isomorphic structures within a graph by mapping vertices to one another while preserving adjacency. In isomorphism testing, symmetry can indicate possible mappings of vertices in one graph to vertices in another, thereby narrowing down the isomorphic candidates. For complex graphs, certain symmetry properties can reveal structural similarities or differences that help detect isomorphism more efficiently.

4. Edge Mapping Methods in Large Graphs: For larger graphs, methods such as canonical labeling or graph hashing help establish unique identifiers based on graph structure. These approaches make it easier to check whether two large graphs could be isomorphic by reducing the problem to a comparison of these labels. Algorithms like Weisfeiler-Lehman also aid in refining candidate vertex mappings by iteratively examining neighborhoods.

5. Directed vs. Undirected Graph Isomorphism: For directed graphs, isomorphism requires that both the structure and the direction of edges align. In undirected graphs, edge direction does not matter. This means that directed graph isomorphism is often more restrictive and computationally challenging due to the additional constraint of matching edge direction between graphs.

Would you like more details or have any questions on these methods? Here are some related questions for further exploration:

1. How does the Weisfeiler-Lehman test refine isomorphism detection for non-isomorphic graphs?
2. What is the importance of automorphism groups in graph theory for identifying isomorphisms?
3. Can spectral graph theory (using eigenvalues of adjacency matrices) help determine non-isomorphism?
4. How does canonical form generation simplify large-scale graph isomorphism tests?
5. What role do graph invariants (such as diameter, girth) play in detecting non-isomorphic graphs?

Tip: For large and complex graphs, always start with basic invariants (like vertex and edge counts) before using more computationally intensive methods.