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Topological indices are often used to predict the physicochemical properties of molecules. The multiplicative sum Zagreb index is one of the multiplicative versions of the Zagreb indices, which belong to the class of most-examined topological indices. For a graph G with edge set E = { e 1 , e 2 , ⋯ , e m } , its multiplicative sum Zagreb index is defined as the product of the numbers D (e 1) , D (e 2) , ⋯ , D (e m) , where D (e i) is the sum of the degrees of the end vertices of e i . A chemical tree is a tree of maximum degree at most 4. In this research work, graphs possessing the maximum multiplicative sum Zagreb index are determined from the class of chemical trees with a given order and fixed number of segments. The values of the multiplicative sum Zagreb index of the obtained extremal trees are also obtained. [ABSTRACT FROM AUTHOR]

Based on the higher order Randié index, we propose three other high order connectivity indices, and obtain the calculation formulas for the new connectivity indices of some chemical trees. [ABSTRACT FROM AUTHOR]

The Wiener polarity index Wp of a graph is defined as the number of unordered pairs of its vertices at distance 3. The problem of finding trees attaining the maximum Wp value, among all chemical trees of a fixed order n, was solved in the paper (Mol. Inf. 2019, 38, 1800076) for n ≥ 8. Motivated by the usage of Wp in a recent publication (J. Chem. Inf. Model. 2020, 60, 1224–1234), in this article we extend the work done in the aforementioned paper by giving a further ordering of chemical trees with respect to the maximum value of Wp. More precisely, we characterize the trees having the second maximum Wp value (which is 3n − 16) from the class of all chemical trees of a fixed order n, for n ≥ 9. [ABSTRACT FROM AUTHOR]

A chemical tree is a tree in which no vertex has a degree greater than four. Two trees T1 and T2 of same orders p, are said to be consecutive trees with respect to the energy, if there exists no tree T of order p satisfying E(T1) < E(T) < E(T2). In this paper the author gives the consecutive chemical trees with respect to energies with edge independence number, denoted by t p(i) where i is the edge independence number and i = 2, 3 and p is the number of vertices. And give the table listing all the possible energy consecutive chemical trees tp(i), i = 2, 3, their polynomials and energies. [ABSTRACT FROM AUTHOR]

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 energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. Two graphs, G1 and G2, are said to be equienergetic if E(G1) = E(G2). We report here the results of the search for pairs of equienergetic acyclic molecular graphs (chemical trees) with the same number n of vertices. There are very few such pairs. The smallest has n = 9 and pertains to 3,3-diethylpentane and 3-methyloctane. Among the chemical trees with n < 18, only five more such pairs and a triplet were found.

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. Two graphs, G 1 and G 2, are said to be equienergetic if E(G 1) = E(G 2).We report here the results of the search for pairs of equienergetic acyclic molecular graphs (chemical trees) with the same number n of vertices. There are very few such pairs. The smallest has n = 9 and pertains to 3,3-diethylpentane and 3-methyloctane. Among the chemical trees with n ≤ 18, only five more such pairs and a triplet were found.

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: Molecular branching is a very important notion, because it influences many physicochemical properties of chemical compounds. However, there is no consensus on how to measure branching. Nevertheless two requirements seem to be obvious: star is the most branched graph and path is the least branched graph. Every measure of branching should have these two graphs as extremal graphs. In this paper we restrict our attention to chemical trees (i.e. simple connected graphs with maximal degree at most 4), hence we have only one requirement that the path be an extremal graph. Here, we show that the generalized Randić index is a suitable measure for branching if and only if where is the solution of the equation in the interval and is the positive solution of the equation . These results include the solution of the problem proposed by Clark and Gutman. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n vertices and let λ1, λ2, . . . , λ n be the eigenvalues of its adjacency matrix. The Estrada index of G is a recently introduced molecular structure descriptor, defined as $${EE (G) = \sum_{i = 1}^n e^{\lambda_i}}$$, proposed as a measure of branching in alkanes. In order to support this proposal, we prove that among the trees with fixed maximum degree Δ, the broom B n,Δ, consisting of a star SΔ+1 and a path of length n−Δ−1 attached to an arbitrary pendent vertex of the star, is the unique tree which minimizes even spectral moments and the Estrada index, and then show the relation EE( S n) = EE( B n, n−1) > EE( B n, n−2) > . . . > EE( B n,3) > EE( B n,2) = EE( P n). We also determine the trees with minimum Estrada index among the trees with perfect matching and maximum degree Δ. On the other hand, we strengthen a conjecture of Gutman et al. [Z. Naturforsch. 62a (2007), 495] that the Volkmann trees have maximal Estrada index among the trees with fixed maximum degree Δ, by conjecturing that the Volkmann trees also have maximal even spectral moments of any order. As a first step in this direction, we characterize the starlike trees which maximize even spectral moments and the Estrada index. [ABSTRACT FROM AUTHOR]

Abstract: A tree is a chemical tree if its maximum degree is at most 4. Hansen and Mélot [P. Hansen, H. Mélot, Variable neighborhood search for extremal graphs 6: analyzing bounds for the connectivity index, J. Chem. Inf. Comput. Sci. 43 (2003) 1–14], Li and Shi [X. Li, Y.T. Shi, Corrections of proofs for Hansen and Mélot’s two theorems, Discrete Appl. Math., 155 (2007) 2365–2370] investigated extremal Randić indices of the chemical trees of order with pendants. In their papers, they obtained that an upper bound for Randić index is . This upper bound is sharp for but not for . In this paper, we find the maximum Randić index for . Examples of chemical trees corresponding to the maximum Randić indices are also constructed. [Copyright &y& Elsevier]

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]

The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree. [ABSTRACT FROM AUTHOR]

Abstract: We study the behavior of the Randić index subject to the operation on a tree T which creates a new tree by deleting an edge ax of T and adding a new edge incident to either a or x. Let be the smallest poset containing all pairs such that and (where is the collection of trees with n vertices and of maximum degree 4). We will determine the maximal and minimal elements of . We present an algorithm to construct -monotone chains of trees such that . As a corollary of our results, we present a new method to calculate the first values of on . [Copyright &y& Elsevier]

• Bounds for arithmetic-geometric index. • Relations between geometric-arithmetic and arithmetic-geometric indices. • Conjectures on graphs that minimize (or maximize) arithmetic-geometric index and various combinations of geometric-arithmetic and arithmetic-geometric indices. The arithmetic–geometric index (AG (G)) was recently introduced as a modification of the well-known geometric–arithmetic index (GA (G)). This paper reports results on searching for extremal AG -graphs for various classes of simple graphs. Additionally, relations between these two indices are elaborated. Results on combinations A G + G A , A G − G A , AG · GA , and AG / GA are given. The paper is concluded with four conjectures that have been derived based on computer investigations. [ABSTRACT FROM AUTHOR]

• Extremal values of vertex-degree based topological indices over chemical trees. • Unification of results in extremal values of VDB topological indices over chemical trees. • Maximal subtree operations to find extremal values of VDB topological indices. • Exponential VDB topological indices and its extremal values. One important topic in chemical graph theory is the extremal value problem of vertex-degree-based topological indices over chemical trees. It is our main goal in this paper to unify many of these results in one general theorem which captures the common properties which are essential. We also apply our results to obtain extremal values of exponential vertex-degree-based topological indices over chemical trees. [ABSTRACT FROM AUTHOR]