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This article is devoted to establishing some extremal results with respect to the difference of two well-known bond incident degree indices [atom-bond connectivity ( ABC) index and Randić ( R) index] for the chemical graphs representing alkanes. More precisely, the first three extremal trees with respect to ABC - R are characterized among all n-vertex binary trees (the trees with maximum degree at most 3). The n-vertex chemical trees (the trees with maximum degree at most 4) having the first three maximum ABC - R values are also determined. [ABSTRACT FROM AUTHOR]

The hyper-Wiener index WW of a chemical tree T is defined as the sum of the products n1n2, over all pairs u, u of vertices of T, where n1 and n2 are the number of vertices of T, lying on the two sides of the path which connects u and u. We examine a slight modification WWW of the hyper-Wiener index, defined as the sum of the products n1n2n3, over all pairs u, u of vertices of T, where n3 is the number of vertices of T, lying between u and u. It is found that WWW correlates significantly better with various physico-chemical properties of alkanes than WW. Lower and upper bounds for WWW, and an approximate relation between WWW and WW are obtained.

The hyper-Wiener index WWW of a chemical tree T is defined as the sum of the product n1 n2 n3, over all pairs u, u of vertices of T, where n1 and n2 are the number of vertices of T, lying on the two sides of the path which connects u and u, and n3 is the number of vertices lying between u and u. An expression enabling the calculation of WWW from the Laplacian eigenvalues of T has been deduced.

Let n1(e|T) and n2(e|T) denote the number of vertices of a tree T, lying on the two sides of the edge e. Let T1 and T2 be two trees with equal number of vertices, let e be an edge of T1 and f an edge of T2. Then e and f are said to be equiseparable if either n1(e|T1) = n1(e|T2) or n1(e|T1) = n2(e|T2). If all edges of T1 and T2 can be chosen so as to form equiseparable pairs, then T1 and T2 are equiseparable trees. A number of molecular structure-descriptors of equiseparable chemical trees coincide, implying that the corresponding alkane isomers must have similar physico-chemical properties. It is shown how equiseparable chemical trees can be constructed in a systematic manner.

Let n1(e|T) and n2(e|T) denote the number of vertices of a tree T, lying on the two sides of the edge e. Let T1 and T2 be two trees with equal number of vertices, let e be an edge of T1 and f an edge of T2. Then e and f are said to be equiseparable if either n1(e|T1) = n1(f|T2) or n1(e|T1) = n2(f|T2). If all edges of T1 and T2 can be chosen so as to form equiseparable pairs, then T1 and T2 are equiseparable trees. A number of molecular structure-descriptors of equiseparable chemical trees coincide, implying that the corresponding alkane isomers must have similar physico-chemical properties. It is shown how equiseparable chemical trees can be constructed in a systematic manner. .

The hyper-Wiener index WW of a chemical tree T is defined as the sum of the products n1n2, over all pairs υ,ν of vertices of T, where n1 and n2 are the number of vertices of T, lying on the two sides of the path which connects υ and ν. We examine a slight modification WWW of the hyper-Wiener index defined as the sum of the products n1n2n3, over all pairs υ,ν of vertices of T, where n3 is the number of vertices of T, lying between υ and ν. It is found that WWW correlates significantly better with various physico-chemical properties of alkanes than WW. Lower and upper bounds for WWW, and an approximate relation between WWW and WW are obtained.

The hyper-Wiener index WWW of a chemical tree T is defined as the sum of the product n1 n2 n3, over all pairs υ,ν of vertices of T, where n1 and n2 are the number of vertices of T, lying on the two sides of the path which connects υ and ν, and n3 is the number of vertices lying between υ and ν. An expression enabling the calculation of WWW from the Laplacian eigenvalues of T has been deduced.

Abstract: The recently introduced atom–bond connectivity (ABC) index has been applied up to now to study the stability of alkanes and the strain energy of cycloalkanes. Here, mathematical properties of the ABC index of trees are studied. Chemical trees with extremal ABC values are found. In addition, it has been proven that, among all trees, the star tree, , has the maximal ABC value. [Copyright &y& Elsevier]