I would appreciate some guidance regarding the concept of matroids in a graph. I have the following graph: V = {1,2,3,4,5,6,7} with E = {(1,2),(1,3,(2,3),(3,4),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)}. The first question is giving the number of elements of the basis of the graph. The way I understood the basis of a matroid is the maximum independent set of F, and in the graph above we can only have a maximum independent set of only 2 Edges. Does this imply that the number of elements of the basis of the graph is 2?
Further, I have to state the number of different basis the graph can have. This question I am really not sure how to go about.
Appreciate all guidance.