In this paper, we compare the (Formula presented.) -numbers and the degree of the (Formula presented.) -polynomials associated with edge ideals of connected graphs. We prove that the (Formula presented.) -number can be arbitrarily larger or smaller than the degree of the (Formula presented.) -polynomial for the edge ideal of a connected graph. We also establish that for any pair of positive integers (Formula presented.) with (Formula presented.), there exists a connected graph (Formula presented.) with the (Formula presented.) -number equal to (Formula presented.) and the degree of (Formula presented.) -polynomial equal to (Formula presented.). Additionally, we show that the sum of the (Formula presented.) -number and the degree of the (Formula presented.) -polynomial is bounded above by (Formula presented.), the number of vertices of (Formula presented.), and we classify all graphs for which this sum is exactly (Formula presented.). Finally, we show that all 13 possible inequalities among the three invariants, the (Formula presented.) -number, the degree of the (Formula presented.) -polynomial, and the Castelnuovo–Mumford regularity can occur in the case of edge ideals of connected graphs. Many of these examples rely on a minimal example of a graph whose v-number is more than the degree of its (Formula presented.) -polynomial. Using a computer search, we show that there are exactly two such graphs on 11 vertices and 25 edges, and no smaller example on fewer vertices, or 11 vertices and less than 25 edges.