Search

The problem of complete characterization of trees with minimal atom-bond-connectivity index (minimal-ABC trees) is a long-standing open problem. Here firstly, we give an affirmative answer to the conjecture, which states that enough large minimal-ABC trees are comprised solely of a root vertex and so-called D z - and D z + 1 -branches. Based on the presented theoretical results here and some already known results, we obtain enough constraints to reduce the search space and solve the optimization problem, and thus, determine exactly the minimal-ABC trees of a given arbitrary order. [ABSTRACT FROM AUTHOR]

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 ∑ u v ∈ E d ( u ) + d ( v ) − 2 d ( u ) d ( v ) , where d ( u ) is the degree of vertex u in G and E 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. [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 atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. Several structural properties of the trees with minimal ABC index were proved recently, however the complete characterization of the minimal-ABC trees is still an open problem. It is known that minimal-ABC trees can have at most one pendent path of length 3. It is also known that the minimal-ABC trees that have a pendent path of length 3 do not contain so-called B k -branches, with k ≥ 4, and do not contain more than two B 2 -branches. Here, we improve the latter result by showing that minimal-ABC trees of order larger than 168 and with a pendent path of length 3 do not contain B 2 -branches. Moreover, we show that trees with minimal ABC index with a pendent path of length 3 do not contain B 1 -branches. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with vertex set V(G) and edge set E(G). The atom-bond connectivity and Randić index of graph G are defined as and , respectively, where du denotes the degree of vertex u in G. Let ABC – R denotes the difference between ABC index and R index. In this paper, we present a further ordering for the ABC – R indices, and determine the fourth, the fifth, the sixth, the seventh and the eighth maximum binary trees with respect to the ABC – R values. [ABSTRACT FROM AUTHOR]

The problem of full determination of trees with a minimal value of the ABC index is very hard and famous in mathematical chemistry. A well-known conjecture is that the big vertices (vertices of degree larger than 2, which are not adjacent to a vertex of degree 2) of a tree with a minimal value of the ABC index induce a star graph. Here we give an affirmative answer to this conjecture and thus make a significant step towards the complete characterization of trees with minimal ABC index. [ABSTRACT FROM AUTHOR]

Let G be a simple, connected and undirected graph. The atom-bond connectivity index ( A B C (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 A B C − R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of A B C − R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for A B C − 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. [ABSTRACT FROM AUTHOR]

Abstract: The atom-bond connectivity (ABC) index of a graph is defined as where is the edge set and is the degree of vertex of . 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. [Copyright &y& Elsevier]