Summary: For natural numbers \(n\) \((n\geq1), r_1, r_2, \dots, r_{3n+1}\), let \(V(F_{n,4}) = \{v_1,v_2,\dots, v_{3n+1}\}\), \(F_{n,4}(r_1, r_2, \dots, r_{3n+1})\) is the new graph that every vertex \(v_i\) in the \(V(F_{n,4}) = \{v_1, v_2, \dots, v_{3n+1}\}\) is bonding \(r_i\) hanging edges. The gracefulness of the graph \(F_{n,4}(r_1, r_2, \dots, r_{3n+1})\) is discussed. It is proved that the graph \(F_{n,4}(r_1, r_2, \dots, r_{3n+1})\) is an alternating graph.