Star graphs

Figure 3. Form factor for a) star graphs numerics (red) versus the power series expansion (15) (green) b) diffusive network (in 2 d) deviations occur for small r, see inset. 

The Markov processes associated with quantum star graphs correspond to systems of weakly coupled edges. Its dynamical properties are determined by the spectrum of the stochastic matrix associated with (14) which is highly degenerate and can be given explicitly (Kottos and Smilansky 1999), that is,... 

Quantum star-graphs have therefore a critical classical spectrum with a spectral gap vanishing proportional to 1 /% one finds indeed spectral statistics intermediate between Poisson and COE statistics. 

A further normalization (binormalization) was also made so as to have an upper limit of the. centric index C equal to one for the case of star-graphs (i.e., graphs consisting of one vertex of degree N-l and N-l verties of degree N) ... 

Figure 5.12 Extreme cases for a graph, containing eight vertices (a) path (linear) graph, (b) star graph. 

The binormalized centric index C is derived by a binormalization, i.e. imposing on all the graphs the same lower bound and an upper bound equal to one for star graphs. In practice, it is obtained from the normalized centric index C divided by its corresponding value for the star graph ... 

TTie reference structure for Pmm is the linear graph, while for / max it is the - star graph, in which all atoms but one are adjacent to a central atom their numerical values are calculated as follows ... 

The binormalized quadratic index was proposed, together with the binormalized centric index C, to provide information on the topological shape of trees, i.e. the trade-off between linear and star graphs. 

The above is a simple example of the anal3dic summation over an alternating series. One can similarly treat more complex paths, which consist of sums of combinations of cyclic paths. In [1], we derive in detail sums over paths (which start and finish at i) constructed on two elemental t3rpes of graphs -star-graphs and chain-graphs. We quote below the results ... 

In the above, Ac, Aq, are primitive cycle functions from which the star-graphs and chain-graphs are constructed. 

Take the current list of primitive graphs, and create a star graph consisting of all of the primitive graphs which contain the current pivot. 

Forming a star graph with all of these graphs pivoting and joining at i we result in Fig. 3. We now consider the two-, three- and four-vertex parts of this star graph. 

Fig. 3. All primitive circuits in the four-vertex graph which pivot at i joined in a star-graph...

Starphene (23) was coined by Clar and Mullen for structures that expanded on the terphenylene motif [32], an all bold connected star graph. Pascal has pushed the envelope on the parent system by preparing the highly distorted perchlorotriphenylene (24) [33], despite common opinion that this molecule could not exist. 

The polycyclic version of the star graph is represented by Diercks and Voll-hardt s triangular [4]phenylene (87), or starphenylene [121]. Some earlier examples are trifoliaphane (88) by Hopf and Psiorz [122], its meta-linked isomer by Vogtle and Kissener [123], and decacyclene (89) for which the crystal structure was only recently reported [124]. 

In the case of k, the structures and index values in Table 3 for A = 7 reveal that information is encoded relating to the degree of star graph-likeness and... 

To illustrate briefly the information in the alpha-modified kappa values, is used as an example. Table 5 shows the values for several molecules. The butyl halides in the list reveal the influence of the covalent radius on the shape modeled by the a parameter on the value. As expected, in bromo-butane is the largest, paralleling a shape that is less star graph-like than the other two halides. The same trends are noted in the cyclic molecules in Table 5. 

Remark 13.7 If is a complete or star graph, all the eigenvalues of its Laplacian are integers. A list of Laplacians with only integer eigenvalues is given in [173]. 

Plots of the concentrations of the inhibitor show the same qualitative behavior. 13.4.2.1 Star Graph... 

The simplest case of a network with a complete node is a star graph. We start our exploration of the nonlinear dynamical behavior of Lengyel-Epstein networks with a 10-node star graph, namely of Fig. 13.3a. Theorem 13.17 implies that F = 1, 10. Choosing r = = 1, we expect from (13.125) that a coupling strength... 

Fig. 13.4 Activator concentrations for the 10-node star graph, for inititil conditions (13.126) (top left), (13.127), (13.128), (13.129), and (13.130) (bottom left). Reprinted from [305]. Copyright 2005, with permission from Elsevier...

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