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), Non-Commuting Graph (Abdollahi, Akbari, and Maimani - Non-commuting graph of a group) 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:
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?
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.
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.