On reciprocal reverse Wiener index

We report some properties, especially bounds for the reciprocal reverse Wiener index of a connected (molecular) graph. We find that the reciprocal reverse Wiener index possesses the minimum values for the complete graph in the class of n-vertex connected graphs and for the star in the class of n-vertex trees, and the maximum values for the complete graph with one edge deleted in the class of n-vertex connected graphs and for the tree obtained by attaching a pendant vertex to a pendant vertex of the star on n − 1 vertices in the class of n-vertex trees. These results are compared with those obtained for the ordinary Wiener index.     

Trees with the minimum Wiener number

The Wiener number (𝒲) of a connected graph is the sum of distances for all pairs of vertices. As a graphical invariant, it has been found extensive application in chemistry. Considering the family of trees with n vertices and a fixed maximum vertex degree, we derive some methods that can strictly reduce 𝒲 by shifting leaves. And then, by a process, we prove that the dendrimer on n vertices is the unique graph reaching the minimum Wiener number. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 331–340, 2000     

Details of the reaction graphs for intramolecular isomerizations of the carboranes

From proposed mechanisms for framework reorganizations of the carboranes C₂B _n-2H _n ,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.     

Maximum energy trees with two maximum degree vertices

The energy E of a graph G is equal to the sum of the absolute values of the eigenvalues of G. In 2005 Lin et al. determined the trees with a given maximum vertex degree Δ and maximum E, that happen to be trees with a single vertex of degree Δ. We now offer a simple proof of this result and, in addition, characterize the maximum energy trees having two vertices of maximum degree Δ.     

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

Structure and some properties of small water clusters

The energies and structures of many small water clusters (H₂O)_n (n=8-26) are calculated using the atom-atom potential functions suggested earlier. For each n, several stable configurations were found that differ in the number of H-bonds and in the topology of the graphs formed by such bonds. The clusters in which the molecules lie at the vertices of convex polyhedra have the lowest-energy but other configurations may have close or even lower energies. For the most stable clusters, the energy dependence on n is close to linear. At 300 K, the mean energies of the clusters behave similarly. Monte-Carlo simulations showed that the clusters undergo pseudomelting at approximately 200 K. Puebla Autonomous University, Puebla, Mexico, Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences. Institute of Mathematical Problems of Biology, Russian Academy of Sciences. Institute of Physical Chemistry, Russian Academy of Sciences Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 6 pp. 113–121, November–December, 1994. Translated by L. Smolina     

On the connectivity index of trees

The connectivity index χ₁(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T(n, r) be the set of trees on n vertices with diameter r. In this paper, we determine all trees in T(n, r) with the largest and the second largest connectivity index. Also, the trees in T(n, r) with the largest and the second largest connectivity index are characterized. Mei Lu is partially supported by NNSFC (No. 10571105).     

The first and second largest Merrifield–Simmons indices of trees with prescribed pendent vertices

The Merrifield–Simmons index of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By T(n,k) we denote the set of trees with n vertices and with k pendent vertices. In this paper, we investigate the Merrifield–Simmons index for a tree T in T(n,k). For all trees in T(n,k), we determined unique trees with the first and second largest Merrifield–Simmons index, respectively.     

Isorpectral multitrees

We examined trees with one multiple edge (of multiplicityk) and report all isospectral graphs found when the number of vertices was n < 9. Tile search for isospectrul multitrees was carried out systematically by constructing the characteristic polynomials of all trees having one weighted edge. For all multitrees havingn < 7 vertices, we tabulated the coefficients of the characteristic polynomial. We restricted the analysis to trees with Me maximal valencyd = 4. The number of graphs considered exceeds 300. The smallest pair of isospectral multitrees (i.e. trees with a multiple edge) hasn = 6 vertices, There is a pair of trees whenn = 7, three pairs whenn = 8, and five pads whenn = 9. In all cases, whenk = I is assumed, isospectral multitrees reduce to the same tree. Whenk = 0 is assume(], isospectral trees produce either the same disconnected graph, or an isospectral forest.Dedicated to Basil E. Gillam, Professor emeritus of the Department of Mathematics and Computer Science at Drake University.Reported in part at the 1986 International Congress of Mathematicians, Berkeley, California, USA.Operated for the U.S. Department of Energy by the Iowa State University under Contract No. W-7405-Eng-82. This work was supported in part by the Office of R.S. Hansen, Director.     

Maximum Randić index on Trees with <Emphasis Type="Italic">k</Emphasis>-pendant Vertices

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. Let T be a tree with n vertices and k pendant vertices. In this paper, we give a sharp upper bound on Randić index of T.     

Characterization of molecular complexity

We put forward a novel index of molecular complexity, ξ, taking into account the symmetry of a molecular graph and the specificity of structural components considered. The ξ index is defined as the sum of augmented valences of all mutually nonequivalent vertices in a molecular graph. The augmented valence of a vertex in a graph is the sum of its valence and valences of all neighboring vertices with the weight 1/2^d depending on their distance, d, from the vertex. The ξ index is examined on the set of octane isomers and some special classes of graphs. It is also compared with a certain number of alternative complexity measures considered in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Computer generation of acyclic graphs based on local vertex invariants and topological indices. Derived canonical labelling and coding of trees and alkanes

A new procedure (GENLOIS) is presented for generating trees or certain classes of trees such as 4-trees (graphs representing alkanes), identity trees, homeomorphical irreducible trees, rooted trees, trees labelled on a certain vertex (primary, secondary, tertiary, etc.). The present method differs from previous procedures by differentiating among the vertices of a given parent graph by means of local vertex invariants (LOVIs). New graphs are efficiently generated by adding points and/or edges only to non-equivalent vertices of the parent graph. Redundant generation of graphs is minimized and checked by means of highly discriminating, recently devised topological indices based either on LOVIs or on the information content of LOVIs. All trees onN + 1 (N + 1 < 17) points could thus be generated from the complete set of trees onN points. A unique cooperative labelling for trees results as a consequence of the generation scheme. This labelling can be translated into a code for which canonical rules were recently stated by A.T. Balaban. This coding appears to be one of the best procedures for encoding, retrieving or ordering the molecular structure of trees (or alkanes).Dedicated to Professor Alexandru T. Balaban on the occasion of his 60th anniversary.