Dumbbell Graphs with Extremal (Reverse) Cover Cost

dumbbell graph B_n(p,l,q) is a bicyclic graph of order n consisting of two vertex-disjoint cycle C_p and C_q with a path of length l-1 jointing them. In this paper, explicit formulas of the (reverse) cover costs, the additive degree-Kirchhoff index (resp. multiplicative degree-Kirchhoff index) in terms of the effective resistance-based invariants and distance-based invariants for dumbbell graphs is displayed. The corresponding extremal graphs for the minimum and maximum cover cost and reverse cover cost of a vertex in a dumbbell graph B_n(p,l,q) are determined, and the extremal graphs for the minimum and maximum additive degree-Kirchhoff index (resp. multiplicative degree-Kirchhoff index) of dumbbell graphs B_n(p,l,q) are also characterized.