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.
Degree of $h$-polynomials of edge ideals
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In this paper, we investigate the degree of $h$-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the $h$-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this marks the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo-Mumford regularity and the degree of the $h$-polynomial of an edge ideal reaches its maximum value, which is the number of vertices in the graph.
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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.