I don't know if it is suitable for MathOverflow, if not please direct it to suitable sites.


I don't understand the following:


I find that there are many ways a graph is associated with an algebraic structure, namely Zero divisor graph ([Anderson and Livingston - The zero-divisor graph of a commutative ring](https://www.sciencedirect.com/science/article/pii/S0021869398978401)), Non-Commuting Graph ([Abdollahi, Akbari, and Maimani - Non-commuting graph of a group](https://www.sciencedirect.com/science/article/pii/S002186930600113X)) and many others.


All these papers receive hundreds of citations which means many people work in this field.


I read the papers, it basically tries to find the properties of the associated graph from the algebraic structure namely when it is connected, complete, planar, girth etc.


My questions are:


1. **We already have a list of many unsolved problems in Abstract Algebra and Graph Theory, why do we mix the two topics in order to get more problems?**
2. It is evident that if we just associate a graph with an algebraic structure then it is going to give us new problems like finding the structure of the graph because we just have a new graph.
**Are we able to solve any existing problems in group theory or ring theory by associating a suitable graph structure? Unfortunately I could not find that in any of the papers.**
3. **Can someone show me by giving an example of a problem in group theory or ring theory which can be solved by associating a suitable graph structure?**


For example suppose I take the ring $(\mathbb Z\_n,+,.)$, i.e. the ring of integers modulo $n$. What unsolved problems about $\mathbb Z\_n$ can we solve by associating the zero divisor graph to it?


**NOTE:** I got some answers/comments where people said that we study those graphs because we are curious and find them interesting.
I am not sure if this is how mathematics works.
Every subject developed because it had certain motivation.
So I don't think this reason that "**Mathematicians are curious about it, so they study it**" stands.
As a matter of fact, I am looking for that reason why people study this field of Algebraic Graph Theory.



Why do we associate a graph to a ring?
The following answer basically involves things I learned about from others in a conversation on the topic of this question, which I have heard voiced many times and is a reasonable question. The paper [ANISOTROPIC GROUPS OF TYPE $A\_n$ AND THE COMMUTING GRAPH OF FINITE SIMPLE GROUPS
by Yoav Segev and Gary M. Seitz](https://msp.org/pjm/2002/202-1/p07.xhtml) uses the commuting graph of a finite simple group to make progress on the Margulis-Platanov conjecture. This earlier Annals paper of Segev [On finite homomorphic images of the multiplicative group of a division algebra](http://emis.impa.br/EMIS/journals/Annals/149_1/7.html)  also uses commuting graphs to solve known conjectures. Of course the graph is not the only thing used and I am not enough of an expert on the subject to say how central the graphs are to the paper.


So the commuting graph of a finite group definitely came up naturally. I am unaware of similar ring theoretic examples. I also am unaware of how planarity and other graph theoretic properties of such graphs play a role in any applications.


Let me also note the nice paper of Peter Cameron [The power graph of a finite group, II](https://www.degruyter.com/view/journals/jgth/13/6/article-p779.xml) which shows that groups with isomorphic power graphs have the same number of elements of each order. This wasn't motivated from outside the theory but I think is still a sign the graph is relevant.


