For edge ideals of connected graphs, the v-number can be arbitrarily larger or smaller than the degree of the h-polynomial, their sum is at most the number of vertices, and equality holds exactly for disjoint unions of star graphs.
The v-numbers of Stanley-Reisner ideals from the viewpoint of Alexander dual complexes
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abstract
We express the v-number of the Stanley-Reisner ideal in terms of its Alexander dual complex and prove that the v-number of a cover ideal is just two less than the initial degree of the its syzygy module. We give some relation between the v-number of the Stanley-Reisner ideal and the Serre-depth of the quotient ring of the second symbolic power of the Stanley-Reisner ideal of its Alexander dual. We also show that the v-number of the Stanley-Reisner ideal of a 2-pure simplicial complex is equal to the dimension of its Stanley-Reisner ring.
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Comparing the $\mathrm{v}$-number and $h$-polynomials of edge ideals
For edge ideals of connected graphs, the v-number can be arbitrarily larger or smaller than the degree of the h-polynomial, their sum is at most the number of vertices, and equality holds exactly for disjoint unions of star graphs.