You can look into my book, titled " A tryst with Matrices: Matrix Shell Model formalism ", freely available for access on researchgate, it presents a new way of visualisation of a complex square matrix of arbitrary order and involves a myriad of interesting mathematical structures, the formalism is very new and still in its infancy, you can work on developing the formalism further, as a part of your doctoral studies.

Ask your advisor!!

Dear Dilara Erdemir

There are promising results may be found by following the second neighborhood conjecture. Indeed it is interesting.

Best regards

Check this post for some topics in graph theory:

https://www.researchgate.net/post/What_are_the_growing_topics_in_graph_theory_for_research

My article "Monomial invariants applied to graph coloring" (posted to the arXiv) explores the interplay between commutative algebra and graph theory. Perhaps, it can serve as a gateway to your topic of interest.

Dear Dilara Erdemir ,

there should be no new research on algebra until the old one (ZFC) is completely destroyed (as well as any use of it).

Could there be a new one after the reform takes charge?

Look at the fundamental definations on algebra: element, group, ring and field. Nothing of that will stay unambiguous.

Inside the group (also: field as the set) there should exist (among themselves) different elements. Could there be any pear inside the set of apples?

Combination (of two elements) inside the set should produce a new element. But a new element only would come by combination outside the set, producing a new set (length by width => area). Old math depends on subsets (= numbers). Incommensurability isn´t used well (the fractions (includung the naturals) can´t be represented by the same ordinate as (for example) the measures π, √2.

Study the reform at: www.mathe-neu.de and present your arguments. Physics takes charge; math gets paramounted to a natural-science. That would be work enough.

A 2,500 years old problem (the binomials of the Babyloniens) gets solved.

A 250 years old problem (the imaginary unit) gets solved.

A 150 years old problem (the second diagonal argument of Cantor) gets solved.

Regards, Peter