Using Castelnuovo-Mumford regularity in Machine Learning research part3 | by Monodeep Mukherjee | May, 2024

The v-number and Castelnuovo-Mumford regularity of cover beliefs of graphs

Authors: Kamalesh Saha

Abstract: The v-number of a graded preferrred I⊆R, denoted by v(I), is the minimal diploma of a polynomial f for which I:f is a major preferrred. Jaramillo and Villarreal (J Combin Concept Ser A 177:105310, 2021) studied the v-number of edge beliefs. On this paper, we analysis the v-number of the quilt preferrred J(G) of a graph G. The first consequence reveals that v(J(G))≤reg(R/J(G)) for any straightforward graph G, which is form of gorgeous because of, for the case of edge beliefs, this inequality doesn’t keep. Our principal consequence relates v(J(G)) with the Cohen-Macaulay property of R/I(G). We provide an infinite class of associated graphs, which fulfill v(J(G))=reg(R/J(G)). Moreover, we current that for every constructive integer okay, there exists a associated graph Gk such that reg(R/J(Gk))−v(J(Gk))=okay. Moreover, we explicitly compute the v-number of cover beliefs of cycles