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Abstract. In this paper we present a theoretical analysis in order to establish maximal and minimal vectors with respect to the majorization order of particular subsets of ℜ n . Afterwards we apply these issues to the calculation of bounds for a topological descriptor of a graph known as the second Zagreb index. Finally, we show how our bounds may improve the results obtained in the literature, providing some theoretical and numerical examples.Mathematics subject classification (2010): 05C35, 05C05, 05C50.

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In this article, we investigate several issues related to the use of the index S G = d i d j , known as the Zagreb index (see Gutman and Das, 2004) or "Smetric" (Alderson and Li, 2007). We present some new upper and lower bounds for S G , in terms of the degree sequence of G. Then, we concentrate on trees and prove that in trees with maximum S G the eigenvector ordering is coherent with the degree ordering; that is, degree central vertices are also eigenvector central. This confirms results given in Bonacich (2007). Further, we show that these trees have minimum diameter and maximum spectral radius in the set of trees with a given degree sequence. A simple application to a company organizational network is provided.

There is convergent consensus among scientists that many social, economic and financial phenomena can be described by a network of agents and their interactions. Surprisingly, even though the application fields are quite different, those networks often show a common behaviour. Thus, their topological properties can give useful insights on how the network is structured, which are the most "important" nodes/agents, how the network reacts to new arrivals. Moreover the network, once included into a dynamic context, helps to model many phenomena. Among the topics in which topology and dynamics are the essential tools, we will focus on the diffusion of technologies and fads, the rise of industrial districts, the evolution of financial markets, cooperation and competition, information flows, centrality and prestige.The volume, including recent contributions to the field of network modelling, is based on the communications presented at NET 2006 (Verbania, Italy) and NET 2007 (Urbino, Italy); offers a wide range of recent advances, both theoretical and methodological, that will interest academics as well as practitioners.Theory and applications are nicely integrated: theoretical papers deal with graph theory, game theory, coalitions, dynamics, consumer behavior, segregation models and new contributions to the above mentioned area. The applications cover a wide range: airline transportation, financial markets, work team organization, labour and credit market.The volume can be used as a reference book for graduate and postgraduate courses on Network Theory and Complex Systems in Faculties of Economics, Mathematics, Engineering and Social Sciences. In Part I, the invited tutorials introduce Graph Theory from the theoretical point of view (Marusic) and the possible applications to economics (Battiston). In Part II, the contributions cover local and global interaction, complex behavior, network games, while in Part III they refer to Markov chains and topology. The applications are all placed in Part IV.Fifteen papers have been selected among roughly thirty submitted extended abstracts; each paper has been reviewed by two referees. Space limitations are the main reason why no more papers have been accepted, although many of them were really interesting. v vi Preface