We consider cooperative games with cooperation structure represented by an undirected communication graph, in which several players playing more important role are selected a priori as the main players. We introduce a solution concept for graph games with main players which generalizes the average tree solution for graph games and takes into account that the main players should be rewarded better than the others. We provide its axiomatic characterization for cycle-free graph games and for non-cycle-free graph games with single main player on the subclass of graph games for which the graphs are cycles. We also show that in case when each ordinary player is connected with only one main player the average tree value for graph games with main players can be represented as a two-step distribution procedure, in which first the total rewards from cooperation are distributed among the unions determined by each of the main players together with all ordinary players related to him, and then the total payoff of each union is distributed among its members. As an application of the new solution concept for graph games with main players, we consider an allocation problem for cooperative games with hub and spoke cooperation structure.