Akbar Ali

FURTHER INEQUALITIES BETWEEN VERTEX-DEGREE-BASED TOPOLOGICAL INDICES

Topological indices play an important role in mathematical chemistry especially in the quantitative structure–property relationship and quantitative structure–activity relationship studies. Recent research indicates that the augmented Zagreb index (AZI) possess the best correlating ability among several topological indices to predict the certain physico-chemical properties of particular types of molecules. In the present work, several novel bounds (lower and upper) for the AZI in terms of first geometric–arithmetic index, Randić index, atom-bond connectivity index, sum-connectivity index, modified second Zagreb index and harmonic index are established. Moreover, the geometric–arithmetic index and modified second Zagreb index are also among the well known topological indices. Various relations between these two topological indices and some of the aforementioned indices are also derived.

A note on chemical trees with minimum Wiener polarity index

The Wiener polarity index (usually denoted by Wp) of a graph G is defined as the number of unordered pairs of the vertices of G which are at distance 3. Denote by CTn the family of all n-vertex chemical trees. In a recent paper, Ashrafi and Ghalavand [1] determined the first three minimum Wp values of n-vertex chemical trees for n ≥ 7 and characterized the chemical trees attaining the first two minimum Wp values among all the members of CTn for n ≥ 4. In this note, the chemical trees with the third minimum Wp value are characterized from the graph family CTn for n ≥ 7, and the chemical trees from the family CTn,n ≥ 4, with the first two minimum Wp values are also obtained in an alternative but shorter way.

On the reduced second Zagreb index of trees

The current note is devoted to investigate the trees, which maximize or minimize the reduced second Zagreb index among all n-vertex trees with fixed number of segments. This note also involves development of some results, which may be used to characterize the extremal trees with respect to the aforementioned index among all n-vertex trees having fixed number of branching vertices.

On the extremal graphs with respect to bond incident degree indices

Abstract Many existing degree based topological indices can be classified as bond incident degree (BID) indices, whose general form is BID ( G ) = ∑ u v ∈ E ( G ) f ( d u , d v ) , where u v is the edge connecting vertices u , v of the graph G , E ( G ) is the edge set of G , d u is the degree of a vertex u and f is a non-negative real valued (symmetric) function of d u and d v . Firstly, here an intuitively expected result is proven, which states that an extremal ( n , m ) -graph with respect to the BID index (corresponding to f ) must contain at least one vertex of degree n − 1 if f satisfies certain conditions. It is shown that these certain conditions are satisfied for the general sum-connectivity index (whose special cases are the first Zagreb index and the Hyper Zagreb index), for the general Platt index (whose special cases are the first reformulated Zagreb index and the Platt index) and for the variable sum exdeg index. With help of the aforementioned result of existence of at least one vertex of degree n − 1 and further analysis, graphs with maximum values of the above mentioned BID indices among tree, unicyclic, bicyclic, tricyclic and tetracyclic graphs are characterized. Some of these results are new and the already existing results are proven in a shorter and more unified way.

An alternative but short proof of a result of Zhu and Lu concerning general sum-connectivity index

Recently, Zhu and Lu, [On the general sum-connectivity index of tricyclic graphs, J. Appl. Math. Comput. 51(1) (2016) 177–188] determined the graphs having maximum general sum-connectivity index among all n-vertex tricyclic graphs. In this short note, an alternative but considerable short approach is proposed for determining the aforementioned graphs.

The inverse Wiener polarity index problem for chemical trees

The Wiener polarity number (which, nowadays, known as the Wiener polarity index and usually denoted by Wp) was devised by the chemist Harold Wiener, for predicting the boiling points of alkanes. The index Wp of chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices (carbon atoms) at distance 3. The inverse problems based on some well-known topological indices have already been addressed in the literature. The solution of such inverse problems may be helpful in speeding up the discovery of lead compounds having the desired properties. This paper is devoted to solving a stronger version of the inverse problem based on Wiener polarity index for chemical trees. More precisely, it is proved that for every integer t ∈ {n − 3, n − 2,…,3n − 16, 3n − 15}, n ≥ 6, there exists an n-vertex chemical tree T such that Wp(T) = t.

The Alkanes with Maximum Wiener Polarity Index

The Wiener polarity index (usually denoted by Wp ) of an alkane is the number of unordered pairs of carbon atoms which are separated by three carbon–carbon bonds. This topological index Wp is useful for predicting the boiling points of alkanes. Deng [MATCH Commun. Math. Comput. Chem. 66 (2011) 305] proved that the maximum Wp value among all alkanes, with n carbon atoms, is 3n-15 . The main purpose of present paper is to find all those alkanes with n carbon atoms, which attain the maximum value of Wp .

A Novel/Old Modification of the First Zagreb Index

In the seminal paper [I. Gutman, N. Trinajstić, Chem. Phys. Lett. 1972, 17, 535–538], it was shown that total electron energy ( Eπ ) of any alternant hydrocarbon depends on the sum of the squares of the degrees of the corresponding molecular graph. Nowadays, this sum is known as the first Zagreb index. In the same paper, another molecular descriptor was proved to influence Eπ , but that descriptor was never restudied explicitly. We call this descriptor as modified first Zagreb connection index and denote it by ZC1* . In this paper, chemical applicability of the molecular descriptor ZC1* is tested for the octane isomers. Some basic properties of ZC1* are also established here. Furthermore, the alkanes with maximum and minimum ZC1* values are determined from the class of all alkanes having fixed number of carbon atoms.

On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

The zeroth-order general Randic index (usually denoted by 0R α) and variable sum exdeg index (denoted by SEIa) of a graph G are defined as 0R α(G) =∑v∈V (G)(dv)α and SEIa(G) =∑v∈V (G)dvadv, respect...

Tetracyclic graphs with maximum second Zagreb index: A simple approach

In the chemical graph theory, graph invariants are usually referred to as topological indices. The second Zagreb index (denoted by M2) is one of the most studied topological indices. For n ≥ 5, let 𝕋𝔼𝕋n be the collection of all non-isomorphic connected graphs with n vertices and n + 3 edges (such graphs are known as tetracyclic graphs). Recently, Habibi et al. [Extremal tetracyclic graphs with respect to the first and second Zagreb indices, Trans. on Combin. 5(4) (2016) 35–55.] characterized the graph having maximum M2 value among all members of the collection 𝕋𝔼𝕋n. In this short note, an alternative but relatively simple approach is used for characterizing the aforementioned graph.