The v-number of monomial ideals

We generalize some results of v-number for arbitrary monomial ideals by showing that the v-number of an arbitrary monomial ideal is the same as the v-number of its polarization. We prove that the v-number v(I(G)) of the edge ideal I(G), the induced matching number im(G) and the regularity reg(R/I(G)) of a graph G, satisfy v(I(G))?im(G)?reg(R/I(G)), where G is either a bipartite graph, or a (C4,C5)-free vertex decomposable graph, or a whisker graph. There is an open problem in \cite{v}, whether v(I)?reg(R/I)+1 for any square-free monomial ideal I. We show that v(I(G))>reg(R/I(G))+1, for a disconnected graph G. We derive some inequalities of v-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v(I(G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg(R/I(G)) can be arbitrarily larger than v(I(G)). Also, we try to see how the v-number is related to the Cohen-Macaulay property of square-free monomial ideals