Abstract
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|suchthatS⊆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.
Highlights
The boom that graph theory has had as an object of study has boosted its application in different contexts and its use as a form of mathematical modeling has become common.The increasing practice of modeling with networks or graphs when it is desired to represent the interactions between elements of a discrete system has made the study of graphs more and more attractive in different mathematical contexts
Let G be a graph of order n, minimum degree δ ≥ 2 and maximum degree ∆
Let G be a graph of order n, size m, maximum degree ∆, and β ∈ (0, δ)
Summary
The boom that graph theory has had as an object of study has boosted its application in different contexts and its use as a form of mathematical modeling has become common. The domination number of a graph G, denoted by γ( G ) is the minimum cardinality of a dominating set of G. Hospitals with such medical equipment will be | B(S)| + |S| − α|S|, i.e., | B(S)| − β|S| with β = α − 1 > −1 This value is known as β-differential of the set S and is denoted by ∂ β (S). In [11], the differential of a set S was considered, in that paper it was denoted by η (S), on the other hand, the minimal differential of an independent set was studied in [12].
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