In this paper we obtain several tight bounds on different types of alliance numbers of a graph, namely (global) defensive alliance number, global offensive alliance number and global dual alliance number. In particular, we investigate the relationship between the alliance numbers of a graph and its algebraic connectivity, its spectral radius, and its Laplacian spectral radius.
Recently, the arithmetic–geometric index (AG) was introduced, inspired by the well-known and studied geometric–arithmetic index (GA). In this work, we obtain new bounds on the arithmetic–geometric index, improving upon some already known bounds. In particular, we show families of graphs where such bounds are attained.
In this work we present numerical results of classical Li\'{e}nard--type systems in a very general context, since we consider several types of derivatives (integer order and fractional order, global and local). Additionally we made theoretical-methodological observations. En este trabajo presentamos resultados num´ericos de sistemas tipo Li´enard en un contexto muy general ya que consideramos varios tipos dederivadas (de orden entero y fraccionario, globales y locales). Adicionalmente hacemos observaciones te ´oricas y metodol´ogicas.
A subset D of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex of D. The total dominating set D is called a total coindependent dominating set if the subgraph induced by V − D is edgeless and has at least one vertex. The minimum cardinality of any total co-independent dominating set is the total co-independent domination number of G and is denoted by γ t,coi (G). In this work we study some complexity and combinatorial properties of γ t,coi (G). Specifically, we prove that deciding whether γ t,coi (G) ≤ k for a given integer k is an NP-complete problem and give several bounds on γ t,coi (G). Also, since any total co-independent dominating set is also a total dominating set, we characterize all the trees having equal total co-independent domination number and total domination number.
In this article, new information dimensions of complex networks are introduced underpinned by fractional order entropies proposed in the literature. This fractional approach of the concept of information dimension is applied to several real and synthetic complex networks, and the achieved results are analyzed and compared with the corresponding ones obtained using classic information dimension based on the Shannon entropy. In addition, we have investigated an extensive classification of the treated complex networks in correspondence with the fractional information dimensions.
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