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In this note, we prove Gluck’s conjecture and Isaacs-Navarro-Wolf Conjecture are true for the solvable groups with disconnected graphsby using Lewis’ group structure theorem with respect to the disconnectedcharacter degree graphs. 1. IntroductionFor a finite group G, we attach a graph ∆(G) to G, whose vertices are primedivisors of degree of some nonlinear irreducible complex character of G, anddistinct vertices p,q are connected when pq divides degree of some irreduciblecharacter of G.By [8, 9], the number of connected components is at most 3 for any finitegroups and 2 for any solvable group. When the graph ∆(G) is disconnected forthe solvable group G, P. Palfy described in [12] the structure of G; furthermoreM. Lewis characterized in [7] the structure of G as follows:Theorem(Lewis). For a solvable group G, the graph ∆(G) is disconnected ifand only if G is one of the groups as Examples 2.1-2.6 in [7].In [7], Examples 2.1-2.6 along with corresponding Lemmas 3.1-3.6 give de-tailed information of structure and character degrees of G, and even the specificgroup examples in individual cases.In this note, we intend to apply the theorem to two open problems ofcharacter theory of finite solvable groups. We write b(G) for the largest ir-reducible character degree of G. Gluck conjectured [3] that the inequality|G : F(G)| ≤ b(G)