On Total and Regular Graphs of a Polynomial

A regular graph of the ring of n × n matrices over a field is a graph whose vertices are nonsingular matrices. Two different matrices are adjacent if their sum is singular. In 2009, S. Akbari, M. Jamaali, and S. Seed Fakhari found that the clique number of this graph is finite whenever the field is not of characteristic 2. The same authors asked if the chromatic number of the graph is finite (for fields of characteristic 0 this question is still open). In this paper, we introduce a concept of total and regular graph of a polynomial, generalizing the regular graph of a matrix ring. We investigate some properties of these graphs and their relationship with the above question. Several new open questions are also posed.