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Let G = (V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∈ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a graph and the order, the size, the girth, the minimum and maximum degree, the diameter, and other domination parameters of the graph.

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]

In this paper, we present a definition of the generalized fractional local derivative, which contains as a particular case several of those reported in the literature. In this framework, we apply this operator to the study of the solutions of some fractional differential equations and we focus mainly on a Gompertz model applied to the study of tuberculosis in Mexico. We also show the effectiveness of this model by fitting it to real data and by comparing it with other models using other well‐known definitions of fractional derivative. [ABSTRACT FROM AUTHOR]

Sustainable water management is important to ensure its availability for future generations. The study of water quality is fundamental for this purpose. Assessing the health of aquatic ecosystems through bioindicators has been shown to be reliable and inexpensive. The objective of this work was to evaluate water quality through a biomathematical model that involves environmental stress indicator organisms and their close relationship with dissolved oxygen. In this direction, a system of differential equations describing the population dynamics of aquatic macroinvertebrates under the influence of dissolved oxygen is proposed. The model is validated by its application in the Coyuca Lagoon, Mexico. Likewise, population changes over time were represented, which allowed us to deduce that the increase or decrease in the aeration/oxygenation rate significantly affects the population dynamics of the bioindicator organisms. In addition, to classify water quality, a one-to-one correspondence was established between water quality and the equilibrium points of the system of differential equations. The results obtained allow inferring that the proposed techniques are useful for the study of water quality since they can predict significant changes in the ecosystem and provide researchers and water managers with tools for decision making. [ABSTRACT FROM AUTHOR]

The inverse degree index of a graph G=(V,E) without isolated vertices is defined as ID(G)=ΣvЄv1/dv, where dv is the degree of the vertex v in G. In this paper, we show a relation between the inverse degree of a graph and the inverse degree indices of the primary subgraphs obtained by a general decomposition of G, we establish some relations between the inverse degree index and other known indices and an application to a specific chemical structure is given. [ABSTRACT FROM AUTHOR]

In this work we obtain several tight bounds for the differential of a graph and we relate the differential of a graph with some known parameters of the graph. [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]

Domination theory is a well-established topic in graph theory, as well as one of the most active research areas. Interest in this area is partly explained by its diversity of applications to real-world problems, such as facility location problems, computer and social networks, monitoring communication, coding theory, and algorithm design, among others. In the last two decades, the functions defined on graphs have attracted the attention of several researchers. The Roman-dominating functions and their variants are one of the main attractions. This paper is a contribution to the Roman domination theory in graphs. In particular, we provide some interesting properties and relationships between one of its variants: the quasi-total Roman domination in graphs. [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]

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity O (n 2) . [ABSTRACT FROM AUTHOR]