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A foliage garden of orchard apple trees can be fun and rewarding, no matter what apple variety is grafted onto root stocks. Depending on variety and climate you can expect to harvest your apples at different times. Once the base skin color of the apples and a pleasing taste convince you that the fruits are harvested mature they require quick ripen off the plant to ensure excellent quality. Then you...
In Chap. 1, we have accounted for the different ways we can enumerate elements in a finite set. In particular, we have mentioned that a permutation $$\Pi : \mathcal{G}\rightarrow \mathcal{G}$$ of the finite set $$\mathcal{G},$$ defines an equivalence relation $$\sim $$ that partitions $$\mathcal{G}$$ into a set of equivalence classes $$\mathcal{G}/ \sim.$$ In the...
A group action is called transitive if it possesses only a single group orbit, so that for every pair of elements a and b, there is a group element $$\gamma,$$ such that $$a = \gamma b.$$
A finite connected undirected graph G(V, E) can be seen as a discrete time dynamical system possessing a finite number of states (nodes) (Prisner 1995). The behavior of such a dynamical system can be studied by means of a transfer operator which describes the time evolution of distributions in phase space. The transfer operator can be represented by a stochastic matrix determining a discrete time...
In the present chapter, we discuss the Markov chains methods of analysis of non-random, connected graphs of a mesoscopic scale which might contain large but finite number of nodes. We use intensively the concept of generalized inverse (Meyer 1975; Campbell and Meyer 1979; Ben-Israel and Greville 2003) which plays an important role in studies of Markov chains, in electrical engineering, linear programming...
In the present chapter, an electrical network is considered as an interconnection of resistors. We demonstrate that random walks defined on connected undirected graphs have a profound connection to electric resistor networks (Doyle and Snell 1984; Tetali 1991; Chandra et al. 1996; Bollobas 1998). In the present chapter, we discuss the effective resistance of electrical networks, the relation between...
The algebraic approach for directed graphs has not been as well developed as for undirected graphs since it is not always possible to define a unique self-adjoint operator on directed graphs.
The structural analysis is an important problem motivating many studies of real – world networks. For instance, the efficient partitioning of a transport network is one of the most effective logistics optimization tools and the biggest opportunity to significantly reduce transportation costs. The structural analysis is also of importance when dealing with extremely large graphs, when we need to cluster...
Graph theory has been developed in regard to its applications in travel, biology, energetics, and many other fields. The general optimization mindset dominating these researches has addressed to graph theory the questions which were often related to finding a shortest path between nodes, as being of the minimum time delay for information transmission and of the minimum cost for connection maintenance...
Geometry and topology of underlying graphs strongly influence on the physical properties of dynamical processes in complex networks. On the one hand, it is interesting to discuss the properties of graphs which might affect the dynamical behavior of a model. On the other hand, the study of large complex networks calls for statistical methods to give an effective description of observed collective dynamical...
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