Proceedings of the 24th Paediatric Rheumatology European Society Congress: Part three

From Springer Nature via Jisc Publications Router.Publication status: PublishedHistory: collection 2017-09, epub 2017-09-0

The largest eigenvalues of adjacency and Laplacian matrices, and ionization potentials of alkanes

893-896If G is a molecular graph and A(G) its adjacency matrix, then the Laplacian matrix is defined as L(G)=D(G)-A(G) where D(G) is the diagonal matrix of vertex degrees. We establish a relation between the largest eigenvalues of A(G) and L(G) in case of the molecular graphs of alkanes. This relation is linear, but differs for alkanes with and without a quaternary carbon atom, revealing the main features of the structure-dependence of their (first) ionization potential. An analogous, yet more general , relation holds for all trees

Ordering of alkane isomers by means of connectivity indices

The connectivity index of an organic molecule whose molecular graph is G is defined as C(l) = S(dudu)1, where du is the degree of the vertex u in G, where the summation goes over all pairs of adjacent vertices of G and where l is a pertinently chosen exponent. The usual value of l is 1/2, in which case c = C(1/2) is referred to as the Randic index. The ordering of isomeric alkanes according to c follows the extent of branching of the carbon-atom skeleton. We now study the ordering of the constitutional isomers of alkanes with 6 through 10 carbon atoms with respect to C( l) for various values of the parameter l. This ordering significantly depends on l. The difference between the orderings with respect to c and with respect to C( l) is measured by a function D and the l-dependence of D was established

Chemical applications of the Laplacian spectrum. VII. Studies of the Wiener and Kirchhoff indices

1272-1278Some further chemical applications of the Laplacian spectra are reported. The Kel'mans theorem for the calculation of the coefficients of the Laplacian characteristic polynomial is stated and exemplified. By means of this theorem a (previously known) formula for the Wiener and Kirchhoff index is deduced. It is shown that the Wiener index is correlated with the "algebraic connectivity", namely, the smallest positive Laplacian eigenvalue. Lower and upper bounds for the Kirchhoff index are obtained

Relations between topological indices of large chemical trees

1241-1245Several approximate relations have recently been established between molecular-graph-based structure descriptors of alkanes, in particular between (a) eigenvalue sum and Hosoya index. (b) greatest graph eigenvalue and connectivity index, (c) Wiener index and smallest positive Laplacian eigenvalue, (d) greatest Laplacian and greatest ordinary graph eigenvalue, (e) Zenkevich and Wiener index, and (f) hyper-Wiener and Wiener index. These all have been found to hold for alkanes with n=10 or fewer carbon atoms, and have verified on samples consisting of all alkane isomers. Applying an algorithm for generating trees uniformly by random we have now tested these regularities for very large chemical trees (n =50). It has been found that regularities (c) and (f) hold equally well in the case of very large chemical trees, whereas regularities (a), (d) and (e) are applicable, but with significantly attenuated accuracy. Regularity (b) vanishes at large values of n.</i

More hyperenergetic molecular graphs

If G is a molecular graph and l1, l2, ..., ln are its eigenvalues, then the energy of G is equal to E(G) = |l1| + |l2| + ... + |ln|. This energy cannot exceed the value . The graph G is said to be hyperenergetic if E(G) >> 2n - 2. We describe the construction of hyperenergetic graphs G for which

Exponent-dependent properties of the connectivity index

457-461The connectivity index is defined as C(λ) =Σ(δuδv)λ, where δv is the degree of the vertex v of the respective molecular graph, and where the summation embraces all pairs of adjacent vertices. The exponent λ is usually chosen to be equal to -0.5 but other options have been considered as well, especially C(-1). We show that whereas C(-0.5) correctly reflects the extent of branching of the carbon-atom skeleton of organic molecules, and is thus a suitable topological index for modelling physico-chemical properties of the respective compounds, this is not the case when the exponent λ assumes larger negative values, in particular when λ= -1.The value of λ is established beyond which C(λ) fails to be a measure of molecular branching

Wiener-type indices and internal molecular energy

In earlier studies it was established that internal molecular energies (Eint) of alkanes can be reproduced, in an approximate yet reliable manner, by means of a molecular-graph-based structure-descriptor U. It was also established that U is linearly correlated with the Wiener index W. We now show that the correlation between U and W is more complicated than earlier expected, and that it cannot be represented by a single line. We also show that a very good linear correlation exists between U and a modified version Wm(l) of the Wiener index, which is thus more suitable for modeling Eint than the ordinary Wiener index

On the number of walks in trees

Dress A, Grünewald S, Gutman I, Lepovic M, Vidovic D. On the number of walks in trees. MATCH Communications in Mathematical and in Computer Chemistry. 2003;(48):63-85.Let W-k be the number of walks of length k in a graph G, and put Delta(k) := Wk+1Wk-1-W-k(2). In recent work, it was shown that exactly one of the following four alternatives holds: Delta(1)greater than or equal to0 and Delta(k)=0 for all k=2,3,... in which case C is said to be harmonic, Delta(2k-10)>0 and Delta(2k)=0 for all k=1,2.... in which case G is said to be almost harmonic, Delta(2k-1)>0 for all k=1,2.... and Delta(2k)>0 for all sufficiently large k in which case G is said to be superharmonic, and Delta(2k-1)>0 for all k=1,2.... and Delta(2k)<0 for all sufficiently large k in which case G is said to be subharmonic. We examined all trees (up to isomorphism) with up to 18 vertices and determined how many of them belong to each of the four classes specified above. In agreement with a previously established result (cf. S. Grunewald, Harmonic Trees, Appl. Math. Lett., to appear) according to which a harmonic tree with at least 3 vertices always has exactly one vertex of degree a(2)-a+1 all of whose neighbours have degree a while all other vertices axe leaves (for some a &ISIN; N-&GE;2), exactly three (with 1, 2, and 7 vertices, respectively) of those trees turned out to be harmonic, no one is almost harmonic, 11 are superharmonic (of which the smallest has 12 vertices), and all others-some 99.994% of all trees examined-are subharmonic

Graph energy-A useful molecular structure-descriptor

1309-1311The energy E(G) of a graph G is, by definition, equal to the sum of absolute values of the eigenvalues of G. The motivation for such a definition comes from molecular orbital theory, where E(G) pertains to the total π-electron energy of the conjugated hydrocarbon whose molecular graph is G. However, E(G) is a well-defined quantity for all graphs.It has been shown that E(G) serves as a structure-descriptor in the case of saturated, σ-electron, systems. The correlations of various physico-chemical properties of alkanes with E(G) are found to be of similar quality as the correlations with other traditionally employed molecular structure-descriptors (Wiener and connectivity indices)