We show that the $$\mathrm {v}$$ -number of an arbitrary monomial ideal is bounded below by its polarization and also find a criteria for equality. By showing additivity associated primes ideals, we obtain v-numbers ideals. prove {v}(I(G))$$ edge I(G), induced matching number {im}(G)$$ regularity {reg}(R/I(G))$$ graph G, satisfy {v}(I(G))\le \mathrm {im}(G)\le , where G either bipartite graph, ...