Let G = (V, E) be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let B(D) be the set of neighbours ofMotivated by an influential maximization problem, in this paper we study the β-differential of G.

If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) ∖ S . Then, the differential of S ∂ ( S ) is defined as | B ( S ) | − | S | . The differential of G, denoted by ∂ ( G ) , is the maximum value of ∂ ( S ) for all subsets S ⊆ V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ∂ ( G ) and ∂ ( Q ( G ) ) . Besides, we exhibit some results relating the differential ∂ ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

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.

<abstract><p>Consider a simple graph $ \Gamma = (V(\Gamma), E(\Gamma)) $ with $ n $ vertices and $ m $ edges. Let $ P $ be a subset of $ V(\Gamma) $ and $ B(P) $ the set of neighbors of $ P $ in $ V(\Gamma)\backslash P $. In the study of graphs, the concept of <italic>differential</italic> refers to a measure of how much the number of edges leaving a set of vertices exceeds the size of that set. Specifically, given a subset $ P $ of vertices, the differential of $ P $, denoted by $ \partial(P) $, is defined as $ |B(P)|-|P| $. The <italic>differential</italic> of $ \Gamma $, denoted by $ \partial(\Gamma) $, is then defined as the maximum differential over all possible subsets of $ V(\Gamma) $. Additionally, the subdivision operator $ {{\mathcal{S}}({\Gamma})} $ is defined as the graph obtained from $ \Gamma $ by inserting a new vertex on each edge of $ \Gamma $. In this paper, we present results for the differential of graphs on the subdivision operator $ {{\mathcal{S}}({\Gamma})} $ where some of these show exact values of $ \partial({{\mathcal{S}}({\Gamma})}) $ if $ \Gamma $ belongs to a classical family of graphs. We obtain bounds for $ \partial({{\mathcal{S}}({\Gamma})}) $ involving invariants of a graph such as order $ n $, size $ m $ and maximum degree $ \Delta $, and we study the realizability of the graph $ \Gamma $ for any value of $ \partial({{\mathcal{S}}({\Gamma})}) $ in the interval $ \left[n-2, \frac{n(n-1)}{2}-n+2\right] $. Moreover, we give a characterization for $ \partial({{\mathcal{S}}({\Gamma})}) $ using the notion of edge star packing.</p></abstract>

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