Lance did a post on Babai's result on Graph Isomorphism (see here). I then did a post asking if Graph Isomorphism has real applications (see here). Lance proofread my post (There were some typos! Really!) and then he was inspired to ask ChatGPT what it thought. The conversation is below (I added some fonts, and my opinion clearly marked, but nothing else). I leave it to the reader to determine what ChatGPT got right and got wrong. In areas that I am unfamiliar with I tend to think that ChatGPT is right, perhaps a case of Gell-man Amnesia (see here).

LANCE: What are some applications of Graph Isomorphism?

ChatGPT:

Isomorphism, a concept from the field of graph theory in mathematics and computer science, refers to the problem of determining whether two graphs are isomorphic, meaning there is a one-to-one correspondence between their vertex sets that preserves adjacency. Despite its theoretical underpinnings, graph isomorphism has practical applications in various domains:

*Chemical Informatics*: It's crucial for identifying chemical compounds. Since chemical structures can be represented as graphs (with atoms as vertices and bonds as edges), determining if two graphs are isomorphic can indicate whether two compounds are structurally identical.

BILL'S OPINION: I suspect simple heuristics work fine. Also I suspect that chemicals are more complicated than graphs. More generally, instead of asking* Does GI have any real applications?* I should have asked* Do the last 45 years of graph isomorphism algorithms produced by our community have any applications? *I suspect the answer is no.

*Pattern Recognition*: In image and pattern recognition tasks, objects can be represented as graphs, and finding isomorphic graphs helps in identifying similar patterns or shapes within different datasets.

BILL'S OPINION: I suspect that in real world pattern recognition, especially for images, approx matches is what you really want. What makes GI hard/interesting is that you need exact Isomorphism. So I doubt GI really helps for Pattern Recognition.

*Network Analysis*: Graph isomorphism is used to analyze and compare different network structures in social networks, biological networks (like neural connections and protein interactions), and the Internet, to understand their underlying properties and dynamics.

BILL'S OPINION: This sounds plausible to me. The people doing the studying are other academics, however, that can still count.

*Computer Vision*: For object recognition and classification, where objects within an image can be represented as graphs, and their identification involves finding isomorphic subgraphs within larger scene graphs.

BILL'S OPINION: Same issue as with Pattern matching.

*Cryptography*: Graph isomorphism problems can serve as the basis for cryptographic protocols, including public key cryptography, due to the computational difficulty of solving graph isomorphism for general cases.

BILL'S OPINION: Just NO. One reason factoring is so useful for crypto is that one can easily generate hard instances. For GI you cannot (at least not yet). I think ChatGPT confused GI being in Zero-Knowledge with GI actually being used for Crypto. More on this later.

*Database Indexing:* In databases, especially graph databases, graph isomorphism helps in query optimization and efficient data retrieval by identifying identical or similar graph structures.

BILL'S OPINION: I can believe this! Gell-man Amnesia?

These applications demonstrate the wide-ranging impact of graph isomorphism beyond theoretical computer science, influencing fields where structural and relational data analysis is crucial.

LANCE: Can you give me specific examples of these?

(BILL: I won't comment on these, except in one case, as it would be the same comments as above.)

ChatGPT:

Certainly! Here are specific examples for the practical applications of graph isomorphism mentioned:

*Chemical Informatics:* A classic example is the identification of isomers in chemistry. Isomers are compounds with the same molecular formula but different structures. Graph isomorphism algorithms can distinguish between different isomers by comparing their graph representations, where vertices represent atoms and edges represent chemical bonds. For instance, butane and isobutane are structural isomers with different connectivity among the same set of atoms, which can be identified as distinct through graph isomorphism.

*Pattern Recognition:* In character recognition systems, letters or symbols can be represented as graphs, where the strokes are edges and intersections are vertices. Graph isomorphism algorithms can then be used to match a scanned, handwritten, or printed character against a database of graph-represented characters to identify the letter or symbol.

*Network Analysis:* In social network analysis, researchers might use graph isomorphism to identify similar subnetworks within larger networks. For example, detecting communities or cliques within social networks that have identical or similar interaction patterns, which can be crucial for understanding social dynamics or the spread of information.

*Computer Vision: *An application in computer vision could be recognizing objects regardless of their orientation in an image. By representing objects as graphs, where vertices correspond to object features and edges to spatial relationships between these features, graph isomorphism can help in identifying an object by finding a match between its graph representation and that of a known object.

*Cryptography*: A specific use case in cryptography is the design of cryptographic primitives based on the Graph Isomorphism problem, such as the Goldreich-Goldwasser-Halevi (GGH) cryptosystem. This cryptosystem constructs a public key from a hard instance of a lattice problem, which is closely related to finding isomorphisms between noisy graph representations, leveraging the computational difficulty of graph isomorphism for security.

BILL: The pointer to the Wikipedia page on the GGH system is here. It has nothing to do with graph isomorphism. So this one is complete garbage.

*Database Indexing*: In the context of graph databases used for storing knowledge graphs or social network data, an example application is the optimization of query processing. When a user queries the database to find data patterns that match a specific graph structure, graph isomorphism algorithms can help quickly locate all instances of the data that are isomorphic to the query graph, enhancing search efficiency and performance.

These examples highlight how graph isomorphism plays a crucial role in solving real-world problems across different domains by leveraging the structure and properties of graphs.

LANCE: Give me a specific research paper in Chemistry that uses graph isomorphism

ChatGPT

A notable research paper that applies graph isomorphism in the field of chemistry is:

Title: "Development and Application of a Novel Graph-Theoretical Approach for Molecular Similarity Analysis"

Authors: Raymond E. Carhart, David H. Smith, Robert Venkataraghavan

Journal: Journal of Chemical Information and Computer Sciences

Year: 1985, Volume: 25, Issue: 1, Pages: 64–73, DOI: 10.1021/ci00045a002

Abstract:

In this paper, the authors introduce a novel method for analyzing molecular similarity based on graph-theoretical concepts. They present an algorithm, known as the MCS (Maximum Common Subgraph) algorithm, which is used to identify the largest subgraph common to two chemical compounds represented as graphs. This process relies on graph isomorphism to determine the structural similarity between molecules. The approach was groundbreaking at the time because it provided a quantitative method for assessing molecular similarity, which is crucial for drug discovery and the development of chemical databases.

The significance of this work lies in its application of graph isomorphism to solve a practical problem in chemistry, demonstrating the interdisciplinary nature of graph theory and its potential to impact research beyond theoretical computer science. This paper has been influential in the fields of cheminformatics and drug design, where molecular similarity plays a key role in identifying potential therapeutic agents.

BILL'S COMMENT: This information from ChatGPT would be more impressive if this paper actually existed.

(ADDED LATER- there is some debate in the comments about if the paper exists.

DOES NOT EXIST: Bill could not find it on the web.

DOES NOT EXIST: Bill went to the website of that journal and did a search and did not find it.

DOES NOT EXIST: The DOI does not point to it.

DOES EXIST: Some slide packet pointed to it as a reference.

If someone leaves a comment with a pointer to it, or emails me a pointer to it, I will update this post.)