Molecular Complexity of Certain Homologous Series of Condensed Benzenoid Hydrocarbons: Limiting Values of the Patency Index and the Index of Spanning-Tree Density

Two indices of molecular complexity, the Patency Index (2017) and the Spanning-Tree Density (2003), are applied to three homologous series of condensed benzenoid hydrocarbons. Calculation of the Spanning-Tree Density requires finding the number of spanning trees in a given molecular graph, which may be achieved by applying the Cycle Theorem (2004) or, in the case of planar graphs in a planar embedding, the theorem of Gutman, Mallion & Essam (1983). To compute the Patency Index, it is necessary to count the number of ladders in the embedded molecular graph. This is done by means of the Dual Cycle Theorem (2017). In the latter, a ladder is conceived of as the edge-set relating to faces as the edge-set of a spanning tree relates to vertices. For a planar graph in a planar embedding, the number of ladders is equal to the number of spanning trees. The three homologous series investigated here are the Linear [n]-Acenes (An), the [n]-Phenacenes (Phn) and the [n]-Helicenes (Hn) (the latter of which are geometrically non-planar but graph-theoretically planar). For these three series, the Spanning-Tree Densities and the Patency Indices may be obtained in closed form and so their behaviour as n → ∞ is easily examined. Though neither index distinguishes between individual members of the three series (An, Phn and Hn) for a specified value of n, this does not preclude the possibility that, within each series, either index may exhibit correlation with physical or chemical data. This work is licensed under a Creative Commons Attribution 4.0 International License

A Dual of the Cycle Theorem and its Application to Molecular Complexity

The duality alluded to in the title is that between the faces and vertices of a graph embedded on a surface. Its recognition in the context of the five Platonic solids is classic. Algebraically, it is present in the equation for Euler’s Polyhedron Theorem and in the various extensions thereof. The Cycle Theorem (CT) establishes a formula for the number of spanning trees contained in a graph embedded on a surface. It is based on the mutual incidences of its cycles (circuits which also carry a sense of direction), i.e., of sub-graphs of the Cn type, endorsed with a sense. These appear (though not exclusively) as the boundaries of faces, so that, so to speak, the Cycle Theorem establishes a result which is essentially about vertices via relations between faces. Among several possible duals of the Cycle Theorem there might thus be one that estab¬lishes a relation which is essentially about faces via relations between vertices. In order to formulate one, we define, for an embedded graph, a feature concerning faces that is dual to a spanning tree. We call it a ladder. A formula is presented for the number of ladders contained in a graph which, in some cases, introduces the concept of ‘artificial vertices’. It is based on the mutual incidences of its vertices. Its form is clearly analogous, or ‘dual’, to the Cycle Theorem formula for spanning trees, previously proposed (in this journal — 2004) by three of the present authors, together with Klein and Sachs. A new index is proposed, which involves ladders. We call it the Patency Index of a graph; its numerical value may be related to molecular complexity. It is effectively the dual of, and is entirely analogous to, the Spanning-Tree Density Index which was earlier (2003) proposed, defined and applied to molecular graphs by one of the present authors and Trinajstić. This work is licensed under a Creative Commons Attribution 4.0 International License

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

What Kirchhoff actually did concerning spanning trees in the course of his classic paper in the 1847 Annalen der Physik und Chemie has, to some extent, long been shrouded in myth in the literature of Graph Theory and Mathematical Chemistry. In this review, Kirchhoff’s manipulation of the equations that arise from application of his two celebrated Laws of electrical circuits — formulated in the middle of the 19th century — is related to 20th- and 21st-century work on the enumeration of spanning trees. It is shown that matrices encountered in an analysis of what Kirchhoff really did include (a) the Kirchhoff (Laplacian, Admittance) matrix, K, that features in the well-known Matrix Tree Theorem, (b) the matrix G encountered in the theorem of Gutman, Mallion & Essam (1983), applicable only to planar graphs, and (c) the analogous matrix M that arises in the Cycle Theorem (Kirby et al. 2004), a theorem that applies to graphs of any genus. It is concluded that Kirchhoff himself was not interested in counting spanning trees, and, accordingly, he did not explicitly do so. Nevertheless, it is shown how the modulus of the determinant of a certain matrix (here denoted by the label C\u27) — associated with the linear equations arising from application of Kirchhoff’s two Laws — is numerically equal to the number of spanning trees in the graph representing the connectivity of the electrical network being studied. Kirchhoff did, however, invoke the concept of spanning trees, introducing them in a complementary fashion by referring to the chords that must be removed from the original graph in order to form such trees. It is further emphasised that, in choosing the cycles in the network being studied, around which to apply his circuit Law, Kirchhoff explicitly selected what would now be called a ‘Fundamental System of Cycles’. This work is licensed under a Creative Commons Attribution 4.0 International License

Badania fizykochemiczne i wytrzymałościowe pyłów z przerobu mas bentonitowych

The investigation results of the physical, chemical and strength properties of dusts from the bentonite sands treatment plants as well as from the reclamation processes of the analogous spent sands, are presented in the paper. The aim of investigations was the determination of preliminary conditions allowing to pelletising dusts in the bowl granulator. The verified examination methods were applied for testing such dust features as: specific density, bulk density of loosely built and of compacted materials as well as related to them porosity, ignition losses and chemical reaction (pH). The grain composition of after reclamation dusts was measured by means of the laser diffraction analysis, allowing to significantly widening the grain size measuring range. The optimal wetting liquid content (in this case: water), at which the water-dust mixture has the best strength properties (formula) was established. Examinations of the moisture content influence on the permeability, compactibility, flowability and friability of samples compacted by means of the standard moulder's rammer were also performed.W niniejszym artykule przedstawiono wyniki badań właściwości fizykochemicznych i wytrzymałościowych pyłów z systemu odpylania przerobu masy z bentonitem oraz systemu regeneracji analogicznej masy zużytej. Badania obejmowały określenie wstępnych warunków umożliwiających grudkowanie pyłów w granulatorze misowym. Zastosowano sprawdzone metody badania takich fizykochemicznych właściwości pyłów jak: gęstość fizyczna (właściwa), gęstość nasypowa luźno usypanego i zagęszczonego materiału i odpowiadająca im porowatość, straty prażenia oraz odczyn chemiczny (pH). Skład granulometryczny pyłów poregeneracyjnych przeprowadzono za pomocą laserowej analizy dyfrakcyjnej, pozwalającej na znaczne rozszerzenie zakresu pomiarowego wielkości cząstek. Określono optymalną zawartość cieczy zwilżającej (w tym przypadku wody), przy której mieszanka wodno-pyłowa posiada najlepsze właściwości wytrzymałościowe [wzór], a także przeprowadzono badania wpływu wilgotności na przepuszczalność, zagęszczalność, płynność zrzutową i osypliwość próbek po zagęszczeniu za pomocą standardowego ubijaka

Used Furan Sand Reclamation in REGMAS Vibratory Unit

The paper, especially dealt with problems of reclamation of used furan sand, carried out in new, vibratory sand reclamation unit REGMAS developed by researches from AGH-University of Science and Technology, Faculty of Foundry Engineering in Cracow (Poland). Functional characteristics of reclamation unit as well as the results of reclamation of used sand with furfuryl resin are discussed in the paper. The quality of reclaim was tested by means of the LOI and pH value, dust content in the reclaim and at least by the the quality of the castings produced in moulds prepared with the use of reclaimed matrix

Regeneracja masy zużytej ze spoiwem nieorganicznym w regeneratorze wibracyjnym REGMAS

The results of the reclamation of spent moulding sands with inorganic binder in the prototype vibratory reclaimer REGMAS - are presented in the paper. Spent moulding sands with modified water-glass, burned as well as not burned, present in a form of agglomerates (lumps), were subjected to the reclamation processes. During individual reclamation cycles the reclaimer efficiency and vibrations amplitudes were determined and physicalchemical and strength investigations were performed for the obtained reclaim materials. The obtained reclaim was used as a component of the matrix of moulding sands with water-glass in the Floster S technology, in which the reclaim constituted 100%, 80% and 50% of the sand matrix - respectively.W artykule przedstawiono wyniki regeneracji zużytej masy formierskiej ze spoiwem nieorganicznym w prototypowym regeneratorze wibracyjnym REGMAS. Procesowi regeneracji poddano zarówno masę zużytą ze szkłem wodnym modyfikowanym przepaloną oraz nieprzepaloną masę zużytą występującą w postaci brył. W trakcie realizacji poszczególnych cykli regeneracji określano wydajność regeneratora oraz amplitudę wibracji, dla których przeprowadzono badania fizykochemiczne oraz wytrzymałościowe otrzymanych regeneratów. Wykonano następujące badania: zawartość pyłów w regeneracie UK , analizę sitową osnowy oraz laserową analizę granulometryczną oddzielonych pyłów, odczyn pH, stratę prażenia oraz zawartość Na2O regeneratu. Uzyskany regenerat użyto jako składnik osnowy mas formierskich ze szkłem wodnym w technologii floster S, w których stanowił odpowiednio 100%, 80% oraz 50% osnowy piaskowej

Assessment of the Possibility of Using Reclaimed Materials for Making Cores by the Blowing Method

The cumulative results of investigations of the possibility of using the reclaimed materials after the mechanical, thermal or mechanical-thermal reclamation for making cores by means of the blowing method in the alkaline CO2 technology, are presented in the paper. Three kinds of spent sands: with furfuryl resin, bentonite and alkaline phenolic resin, obtained from the foundry, were subjected to three kinds of reclamation: mechanical, thermal and combined mechanical-thermal, applying for this aim adequate experimental devices. The obtained reclaims were assessed with regard to the degree of the matrix liberation from the determined binding material. Reclaims of moulding sands with binders of the form of resin were assessed with regard to ignition loss values and pH reaction, while reclaims of moulding sands with bentonite with regard to the residual clay content and pH value. In all cases the results of the performed sieve analyses were estimated and the average characteristic diameter dl was determined. The reclaimed matrix was applied as a full substitute of the fresh high-silica sand in typical procedures of preparing core sands used for making shaped samples for bending strength investigations, Rgu

Reclamation of the Inorganic Spent Sand from the Warm-Box Technology in the REGMAS Reclaimer

The results of investigations of the reclamation of spent inorganic moulding sands from the warm-box technology are presented in the paper. The reclamation was performed in the prototype vibratory reclaimer: REGMAS. Water-glass with some organic additions influencing its knocking out properties was the binder of this spent sand. The following indices were applied for the reclaimability assessment: ignition loss, Na2O content, dusts amounts generated in the reclamation process, pH value of the reclaim as well as strength of moulding sands prepared with the reclaimed material, with the fresh sand and with the mixture: fresh sand-reclaim