Topological Formula Research Articles

Topological evaluation of feedback matrices in multiple-loop feedback amplifiers

Topological formulas for the evaluation of the elements of the return difference and the null return difference matrices of a multiple-loop feedback amplifier are presented. They are expressed in terms of the directed-tree and directed-two-tree admittance products in the associated digraph. In the situation where the controlling parameters of all the controlled sources are chosen as the elements of interest, the associated digraph is simply the graph representing the feedback amplifier after the independent source and controlled sources have been removed. The significance of the present approach is that it not only provides a short-cut for the evaluation of the feedback matrices, but also gives an insight into the behavior of the feedback amplifier under consideration.

A simplification of Stenger's topological degree formula

A formula due to F. Stenger expresses the topological degree of a continuous mapping defined on a polyhedron inR n as a constant times a sum of determinants ofn×n matrices. We replace these determinant evaluations by a scanning procedure which examines their associated matrices and at each ofn?1 steps discards at least half of the matrices remaining from the previous step. Finally we obtain a lower bound for the number of matrices present originally, thus giving an estimate for the minimum amount of labour needed in many cases to compute the degree using this method.

Topological formulation of general linear network functions

This paper first discusses the corrections of the errors which Jong and Zobrist made in Ref. ( 1) and next describes the sign determination suitable forthese topological formulas. The technique for sign determination is based on the node partitioning algorithm which is also presented.

An approximate topological formula for the homo-lumo separation in alternant hydrocarbons

A first order formula is developed to give an approximate estimate of the Hückel separation of the highest occupied and lowest unoccupied molecular orbitals (the HOMO-LUMO separation) in alternant hydrocarbon species. It is found that the formula leads to a value which is too low in most cases. The average deviation for the 24 species considered amounts to 0.116 β. The new formula represents the first purely topological formula for this parameter.

Topological studies on heteroconjugated molecules

A graph-theoretical analysis of certain π-electron properties of alternant molecules with one heteroatom is given. Topological formulas for total π-electron energy, π-electron charge density, bond order and various polarizabilities are derived. The main results of the paper are summarized in Rules 1-7.

A Method for Calculation of Resonance Energy of Benzenoid Hydrocarbons

A semiempirical topological formula (2) is derived, which reproduces Aihara’s resonance energy [1] of benzenoid hydrocarbons with a chemically negligible error of 2 -3%.

New Topological Formula and Rapid Algorithm for Reliability Analysis of Complex Networks

Part I derives a new topological formula for the terminalpair reliability of complex networks. The formula generates only non-cancelling terms. The non-cancelling terms in the reliability expression correspond one-to-one with the acyclic subgraphs of the given probabilistic graph. Part II introduces the concept of neutral sequences in acyclic graphs; several of their important properties are established. Based on these results a powerful algorithm for generating the reliability expression is presented. The reliability expression is obtained in symbolic factored form. Examples indicate that the present algorithm is appreciably faster than earlier methods. The properties of cyclic and acyclic graphs established in this paper are significant new results in the theory of digraphs and have further ramifications and wider application than in reliability.

A new signal flowgraph formulation of symbolic network functions

Symbolic topological analysis of the closed primitive signal flowgraph corresponding to a linear active or passive network having an edge admittance matrix is considered. First, necessary and sufficient conditions for the occurence of term cancellation in Mason's formula for the graph determinant are established. Then, a new signal flowgraph topological formula for the graph determinant in which no term cancellation occurs is proved. A computer program to implement the new formula is described. It is shown that this program is more efficient than existing programs based on Mason's formula.

New Signal Flow Graph Topological Formulas for Linear Active Networks

Topological analysis of the signal flow graph associated with the hybrid system of equations for a linear active or passive electrical network for which the element admittance matrix exists and is diagonal is considered. First, the term cancellation which occurs in Mason's topological formulas is investigated. Necessary and sufficient conditions on the signal flow graph topology such that a term in the expansion of the graph determinant and cofactors either cancels out with another term in the expansion or does not cancel are established. Properties of the associated network which result in non-cancelling terms are given and the number of non-cancelling terms is determined. Second, new signal flow graph topological formulas for the graph determinant and cofactors are proven. These formulas are such that no term cancellation occurs and are readily adaptable to computer implementation. In addition, the number of terms in these formulas is independent of the network tree used to formulate the signal flow graph. Examples are given to illustrate the new formulas.

New topological formula and rapid algorithm for reliability analysis of complex networks : A. Satyanarayana and A. Prabhakar. IEEE Trans. Reliab.R-27, (2) 82 (June 1978)

New topological formula and rapid algorithm for reliability analysis of complex networks : A. Satyanarayana and A. Prabhakar. IEEE Trans. Reliab.R-27, (2) 82 (June 1978)

A Topological Formula for Total π-Electron Energy

Abstract Total π-electron energy E is presented as a sum of first-neighbour interaction terms λ and contributions of cycles ζ [Equation (2)]. The main conclusion of the work is that the total effect of λ's is a constant, independent of the molecular structure. Therefore, E is an additive function of ζ′s only.

Topological Properties of Bond Order

Abstract Integral topological formulas are obtained, relating the bond order with the characteristic poly­ nomials of the molecular graph and its certain subgraphs. Several properties of the bond order are proved.

A class of approximate topological formulas for total π-electron energy

A class of approximate topological formulas, Eqs. (12), (15), (25), and (27) for total π-electron energy E is derived. These formulas reflect the dependence of E on a number of structural details of the molecule: the number of atoms, bonds, rings, and nonbonding MO’s, the ring size, the algebraic structure count, and the branching. The formulas are obtained using analytical considerations and contain no fitting parameters.

An algorithm for direct evaluation of network transfer functions

A method is proposed for computing the network transfer functions by direct evaluation of the determinant and the appropriate cofactors of the node-admittance matrix of the network using rules governed by topological formulae but without actually generating trees of the corresponding graph. This method suggests a quick analysis (and eventually design) of networks avoiding any matrix analysis or matrix inversion techniques, as in usual practice.

Topological studies on heteroconjugated molecules. The stability of alternant systems with one heteroatom

The stability of alternant conjugated molecules, containing one heteroatom, is studied by means of graph theory. The Coulson integral representation of total π-electron energy yields a formula for the difference between the π-energy of a heteroconjugated molecule and its parent hydrocarbon. The integral is calculated approximatively and a simple topological formula is derived, enabling the prediction of the stability of a given heteroconjugated molecule relative to its parent hydrocarbon.

Indefinite-admittance matrix formulation of feedback amplifier theory

It is shown that the return-difference and the null return-difference functions of a feedback amplifier can be elegantly and compactly formulated in terms of the first- and the second-order cofactors of the elements of its indefinite-admittance matrix. The relationship between the generalized cofactors of the elements of the cutset-admittance matrix and the second-order cofactors is indicated. Based on these results, topological formulas for these functions are presented. The significance of the new formulation is that it avoids the necessity of interpreting the loop gain in terms of the actual signal transmission around the closed loops in a flow graph, which in many cases is very misleading.

Topological analysis of linear active networks at a number of selected ports

Explicit topological formulae are obtained to characterize linear active networks at a number of ports. Three different representations based on the impedance, admittance and scattering matrices are obtained. The conditions for the existence of each representation and the relations between them are also given.

The nonexistence of topological formula for total ?-electron energy

A definition of the topological formula for total π-electron energy (E) is given and its necessary analytical from is shown to be a linear combination of graph invariants. It is demonstrated that no exact topological formula for E can exist, i.e. that E cannot be expressed as a function of graph invariants. The proof of the nonexistence of the topological formula is based on Ulam's conjecture.

Topological formulation of symbolic network functions and sensitivity analysis of active networks

A new and efficient algorithm for topological formulation of network functions in symbolic form for a general linear (passive or active) network is presented. This algorithm, which is based on the concept of graph partitioning developed by Dunn and Chan, has all of the advantages of the existing topological formulas, such as overcoming the problem of sign evaluation and greatly reducing the number of term cancellations and the number of topological terms. In addition, and perhaps the most important, it presents a step-by-step procedure for formulating network functions that can readily be adapted for computer implementation. Finally, it is shown that the theoretical results developed herein are most suited for sensitivity analysis of linear active networks.

Mixed-type topological formulas for general linear networks

"Topological formulas" are established for the most general type of linear electrical networks in which the characteristics of branches are expressed in terms partly of impedances and partly of admittances. We shall call this type of topological formulas "mixedtype topological formulas." Moreover, the networks under consideration may contain any type of mutual coupling such as active elements, controlled sources, ideal transformers, etc. The formulas are derived directly from the fundamental laws in network theory. Existing topological formulas are particular cases of those presented in this paper, and our way of derivation will be one of the most straightforward and can be applied to derive any known topological formula. Use is made only of the topological properties of original networks but not of auxiliary graphs such as voltage graphs, current graphs, and (signal-) flow graphs. The denominators and numerators of the formulas are expressed as the sum of the immittance products corresponding to the tree-cotree pairs of special kinds, and the sign of each term in the sum is characterized in topological terms.