zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Further results on atom-bond connectivity index of trees. (English) Zbl 1216.05161

Summary: The atom-bond connectivity (ABC) index of a graph $G$ is defined as

$\text{ABC}\left(G\right)=\sum _{uv\in E\left(G\right)}\sqrt{\frac{{d}_{u}+{d}_{v}-2}{{d}_{u}{d}_{v}}},$

where $E\left(G\right)$ is the edge set and ${d}_{u}$ is the degree of vertex $u$ of $G$. We give the best upper bound for the ABC index of trees with a perfect matching, and characterize the unique extremal tree, which is a molecular tree. We also give upper bounds for the ABC index of trees with fixed number of vertices and maximum degree, and of molecular trees with fixed numbers of vertices and pendent vertices, and characterize the extremal trees.

MSC:
 05C90 Applications of graph theory 05C05 Trees
References:
 [1] Cvetković, D.; Gutman, I.; Trinajstić, N.: Kekulé structures and topology, Chem. phys. Lett. 16, 614-616 (1972) [2] Estrada, E.; Torres, L.; Rodríguez, L.; Gutman, I.: An atom-Bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A, 849-855 (1998) [3] Estrada, E.: Atom-Bond connectivity and the energetic of branched alkanes, Chem. phys. Lett. 463, 422-425 (2008) [4] Furtula, B.; Graovac, A.; Vukičević, D.: Atom-Bond connectivity index of trees, Discrete appl. Math. 157, 2828-2835 (2009) · Zbl 1209.05252 · doi:10.1016/j.dam.2009.03.004 [5] Hansen, P.; Mélot, H.: Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index, J. chem. Inf. comput. Sci. 43, 1-14 (2003) [6] B. Zhou, R. Xing, On atom-bond connectivity index, preprint.