Kirchhoff matrix

Kirchhoff matrix Laplacian matrix Kirchhoff number -> resistance matrix Klopman-Henderson cumulative substructure count... 

Laplacian matrix (L) ( admittance matrix, Kirchhoff matrix)... 

As outlined in Sect. 1, a phantom network is defined as a network with the fictitious property that chains and junctions can move freely through one another without destroying the connectivity of the network. Usually, models of rubber elastic networks are built up from Gaussian chains and the topology of connectivity is completely described by the reduced Kirchhoff matrix of Eq. (6). However, Staverman pointed out that for a network with a given Kirchhoff matrix, the model has to be completed by additional assumptions. 

Molecular architectures can be structurally classified as being more comb-like or Cayley tree-like. Structure has impact on the radius of gyration, which is larger for linear molecules than for branched molecules of the same weight (number of monomer units), since the latter are more compact. The ratio between branched and linear radius is usually described by a contraction factor . Furthermore, Cayley tree-like structures are more compact than comb-like structures [33, 56]. We will show here how to obtain the contraction factor from the architectural information. The squared radius of gyration is expressed in monomer sizes. According to a statistical-mechanical model [55] it follows from the architecture as represented in graph theoretical terms, the KirchhofF matrix, K, which is derived from the incidence matrix, C [33] ... 

Here, n is the number of monomer units and Tr(A ii) denotes the trace of A ij, being the matrix with n — 1 reciprocals of the eigenvalues of the Kirchhoff matrix K. The full n X n sized matrix K is calculated from ... 

If one assumes that all bonds have the same (unit) resistance one can write a resistance-distance matrix 0. This matrix has also been referred to as a Kirchhoff matrix, in view of the fact that it rests on Kirchhoff s current flow laws. The resistance-distance matrix better reflects interatomic distances in cyclic compounds than the ordinary distance matrix, as it takes into account not only the shortest paths in a graph but also the presence of alternative connections between vertices. 

However, for a network with given Kirchhoff matrix three types of phantom networks can be defined free phantom networks, fixed phantom networks and localised phantom networks. 

The properties of such a localised expanded network can be derived from James theory in Flory s version. Consider a large phantom network with variable Kirchhoff matrix T. The configuration function of the free phantom network is Zfree = exp[- J T J ] where R represents the set of N vectors (3N components) R, Rj of the junctions i, j. James and Flory have shown that for a fixed phantom network with a junctions fixed and r junctions free, the configuration function can be written... 

We have already argued that a localised phantom network appears to be the most suitable type of phantom network to correspond with a real network. Once the chemical structure of the network is given, the Kirchhoff matrix is known. The phantom network corresponding to a given real network should have the same Kirchhoff matrix... 

With a given Kirchhoff matrix, a multitude of i -configurations is compatible. Once the equilibrium positions are localised in a definite configuration, S o, all -config-... 

II The R -configuration in the real network may be imcompatible with the Kirchhoff matrix, corresponding to the chemical structure of the network. 

Clearly, this assumption implies that, once the Kirchhoff matrix and the size of the fluctuation domains ate known, also the reai configuration of the real network is known. It also implies that the ),eai configuration is not in equilibrium with the network forces. Therefore, the theory of localised or fo fixed phantom networks is not applicable. 

To networks of class 2 as defined by Eq. (40b) neither the theory of phantom networks with fixed junctions of James nor the theory of localised phantom networks as presented in Sects. 2 and 3 is applicable since no junctions are fixed and the network is not in equilibrium withjtself. There is no direct and simple relation between the Kirchhoff matrix and th i -configuration. The i reai-values have been displaced from the corresponding R ph-values by liquid or entanglement forces. The exact nature of these forces in not known. 

The first course is followed in numerous papers in which the concept of entanglement junctions is introduced. An excellent survey of the literature is given by Graessly With this concept one introduces a Kirchhoff matrix, an / -configuration and a number of elastic degrees of freedom that have no relation with the chemical structure of the network, except that the additional assumption of the additivity of elasticity due to chemical and physical junctions is made. 

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