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Let G = (V , E) be a graph, and let β ∈ R . Motivated by a service coverage maximization problem with limited resources, we study the β -differential of G. The β-differential of G, denoted by ∂ β (G) , is defined as ∂ β (G) : = max { | B (S) | − β | S | s u c h t h a t S ⊆ V } . The case in which β = 1 is known as the differential of G, and hence ∂ β (G) can be considered as a generalization of the differential ∂ (G) of G. In this paper, upper and lower bounds for ∂ β (G) are given in terms of its order | G | , minimum degree δ (G) , maximum degree Δ (G) , among other invariants of G. Likewise, the β -differential for graphs with heavy vertices is studied, extending the set of applications that this concept can have. [ABSTRACT FROM AUTHOR]

Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of γ -sets (minimum dominating sets), the graph is called γ -transitive if given two γ -sets there exists an automorphism which maps one onto the other. We deal with two families: paths P n and cycles C n . Their γ -sets are fully characterized and the action of the automorphism group on the family of γ -sets is fully analyzed. [ABSTRACT FROM AUTHOR]