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The clustering coefficient of a vertex v, of degree at least 2, in a graph Γ is obtained using the formula C (v) = 2 t (v) deg (v) (deg (v) − 1) , where t (v) denotes the number of triangles of the graph containing v as a vertex, and the clustering coefficient of Γ is defined as the average of the clustering coefficient of all vertices of Γ , that is, C (Γ) = 1 | V | ∑ v ∈ V C (v) , where V is the vertex set of the graph. In this paper, we give explicit expressions for the clustering coefficient of corona and lexicographic products, as well as for the Cartesian sum; such expressions are given in terms of the order and size of factors, and the degree and number of triangles of vertices in each factor. [ABSTRACT FROM AUTHOR]

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]