Connected Degree of Fuzzifying Matroids

Abstract

Polya’s plausible reasoning methods are crucial not only in discovery of mathematics results, modeling methods, and data processing methods but also in many practical problems’ solving. This paper exemplifies how to use Polya’s plausible reasoning methods to generalize the popularized notion of 2-connectedness of graphs to a more universal notion of the connected degree of fuzzifying matroids. We introduce the connectedness of fuzzifying matroids, which is generalized from 2-connectedness of graphs, connectedness of matroids, and 2-connectedness of fuzzy graphs. Moreover, the connected degree of fuzzifying matroids is presented by considering the fuzziness degree of connectedness. It is proved that a fuzzifying matroid $\mathbb{M}$ is connected, which is equivalent to its connectedness degree $\text{Con}\left(\mathbb{M}\right)$ is the biggest (i.e., $\text{Con}\left(\mathbb{M}\right)=1$ ). This, together with other properties of the connected degree of fuzzifying matroids, demonstrates the rationality of the proposed notion. Finally, we describe the concepts of this paper through some examples.