thank you for taking the time to read this post.
Some Context on My Background: I am an undergraduate student in mathematics, and I recently coauthored my first paper with my summer mentor. While working on the project, I had the chance to attend a few conferences in the area, and sat in on research seminars and colloquia regularly -- I continued attending seminars, when my schedule allowed, this past fall. For coursework, I've completed "core" subjects of linear algebra, abstract algebra, real analysis, (elementary) number theory, combinatorics + graph theory, and an advanced linear algebra course. My current interests are in algebraic combinatorics, but I'm not 100% sure where I'd want to specialize (and still have a lot to learn), should I get the privilege of joining a PhD program down the road.
I mainly include this section to acknowledge inevitable gaps in my understanding of niche topics, and remain well aware of the "reinventing the wheel" phenomenon in early projects -- Professor Tao's advice on "healthy skepticism" has been helpful.
The Topic -- Brief Overview:Since this post is centered around the research process generally, I'll try to keep my explanation concise, and remain open to any questions/constructive critiques. I'm curious about a class of graphs that resemble a collection of windmill graphs (I've seen various forms of notation for these graphs -- namely $K_n^{(k)}$ and $W_d(n,k)$), all joined at a common "central vertex." Specifically, each of our $n$ windmill graphs, $W_d(m,k)$, are constructed for a fixed integer $k\ge1$ and integers $m=1,2,...,n$, respectively, where $k$ represents the total number of copies of the complete graph $K_m$ in each individual windmill graph.
My motivation for this idea comes from a naive question in linguistics I was thinking about in October, which somewhat "evolved" into something with mathematical structure that I feel more comfortable handling. Initially, I aimed to describe this class of graphs using a recursive style of construction, where edges are weighted based off "distances" from specific vertices, but this method still requires a lot of refinement (I have an algorithm that successfully depicts the weighted graphs with python, but would prefer a formal explanation for study).
I've shared this idea with a few friends and mentors at my university/local universities, and many of them believe the class of graphs could be novel. If this is true, there are some things I'm wondering about:
How does one define and communicate a class of graphs in a meaningful way?What would the structure of the associated paper look like?What graph properties would people be most concerned with at first?
Windmill graphs are well studied -- should I start my literature review by familiarizing myself with results involving windmill graphs? There are so many interesting directions I could see this project going in -- how to choose?
Conducting a Review of the Literature: As noted in one of the questions above, I've begun collecting a few papers involving windmill graphs that could be interesting, but my search has been otherwise haphazard at best. I'm curious how mathematicians narrow their search, especially if the topic of interest is initially quite broad. I also began reviewing Diestel's Graph Theory, to ensure I'm approaching the project with decent and refreshed (general) graph theory knowledge-base; Do others have additional book/resource recommendations in graph theory and related areas?
Mentorship/Talking About My Idea: I know undergraduate students typically have mentors for projects like these... because we often have no idea what we're doing! My summer mentor was great, but I'm not sure this exactly falls into his area of specialty. I also worry that my excited tendency to share the idea, with various mathematicians I meet, will make me seem immature/annoying -- if I don't do it correctly. Thus, I'm wondering:
What's the best way to open up these conversations? Do I need a mentor at this point? If my ideas aren't developed enough for mentorship, what else should I focus on? Who/which specialties might be interested?
Overall Process + Misc.: If I'm able to refine this idea as I hope, how do I organize ideas for -- as mentioned in the literature review portion -- a general class of graphs that one could say possibly many things about; how many of these "directions" should I realistically focus on? (my previous project involved proving a specific conjecture, so less of these specific skills were touched upon -- although I learned a good deal about organizing the smaller pieces of that idea + reading academic papers).
Finally, how do I go about judging the "significance" of this work -- is it even valuable/possible to do at my career stage, or should I be more focused on learning/enjoying the process. Since I'm still building my foundations in math, I probably ought to refrain from dedicating myself to lossy "rabbit-holes," over working on my problem sets, once the spring semester starts up...
I'd appreciate any thoughts you may have. Thank you again! (below are a few figures depicting some variations of the graph, should you be curious)