On structural properties of trees with minimal atom-bond connectivity index II (original) (raw)
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On structural properties of trees with minimal atom-bond connectivity index
Discrete Applied Mathematics, 2014
The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In [14], it was shown that trees with minimal ABC index do not contain so-called B k-branches, with k ≥ 5, and that they do not have more than four B 4-branches. Our main results here reveal that the number of B 1 and B 2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B 1-branches and more than eleven B 2-branches.
Forbidden branches in trees with minimal atom-bond connectivity index
2017
The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph G, the ABC index is defined as ∑_uv∈ E√(d(u) +d(v)-2/d(u)d(v)), where d(u) is the degree of vertex u in G and E(G) denotes the set of edges of G. In this paper we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a B_4-branch and B_1 or B_2-branches.
arXiv (Cornell University), 2017
The atom-bond connectivity (ABC) index is one of the most investigated degreebased molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length 3, with the conjecture that it cannot be a case if the order of a tree is larger than 1178. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length 3 if its order is larger than 415.
Atom–bond connectivity index of trees
Discrete Applied Mathematics, 2009
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, S n , has the maximal ABC value.
Efficient computation of trees with minimal atom-bond connectivity index
Applied Mathematics and Computation, 2013
The atom-bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph G, the ABC index is defined as uv∈E(G) (d(u)+d(v)−2) d(u)d(v) , where d(u) is the degree of vertex u in G and E(G) is the set of edges of G. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal ABC index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of trees with minimal ABC index. The obtained results disprove some existing conjectures and suggest new ones to be set.
On the difference between geometric-arithmetic index and atom-bond connectivity index for trees
Journal of Mathematics and Computer Science
Let G be a simple and connected graph with vertex set V(G) and edge set E(G). The geometric-arithmetic index and atom-bond connectivity index of graph G are defined as GA(G) = uv∈E(G) 2 √ d u d v d u +d v and ABC(G) = uv∈E(G) d u +d v −2 d u d v , respectively, where the summation extends over all edges uv of G, and d u denotes the degree of vertex u in G. Let (GA − ABC)(G) denote the difference between GA and ABC indices of G. In this note, we determine n-vertex binary trees with first three minimum GA − ABC values. We also present a lower bound for GA − ABC index of molecular trees with fixed number of pendant vertices.
Atom-bond connectivity index of graphs
Discrete Applied Mathematics, 2010
The recently introduced atom-bond connectivity (ABC) index has been applied up until now to study the stability of alkanes and the strain energy of cycloalkanes. Furtula et al. (2009) [3] obtained extremal ABC values for chemical trees, and also, it has been shown that the star K 1,n−1 , has the maximal ABC value of trees. In this paper, we present the lower and upper bounds on ABC index of graphs and trees, and characterize graphs for which these bounds are best possible.
The minimal-ABC trees with B1-branches
PloS one, 2018
The atom-bond connectivity index (or, for short, ABC index) is a molecular structure descriptor bridging chemistry to graph theory. It is probably the most studied topological index among all numerical parameters of a graph that characterize its topology. For a given graph G = (V, E), the ABC index of G is defined as [Formula: see text], where di denotes the degree of the vertex i, and ij is the edge incident to the vertices i and j. A combination of physicochemical and the ABC index properties are commonly used to foresee the bioactivity of different chemical composites. Additionally, the applicability of the ABC index in chemical thermodynamics and other areas of chemistry, such as in dendrimer nanostars, benzenoid systems, fluoranthene congeners, and phenylenes is well studied in the literature. While finding of the graphs with the greatest ABC-value is a straightforward assignment, the characterization of the tree(s) with minimal ABC index is a problem largely open and has recen...
Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index
Symmetry
Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(G)) and Randić index (R(G)) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as ABC−R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of ABC−R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for ABC−R index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory.