Minimum Atom-Bond Sum-Connectivity Index of Trees With a Fixed Order and/or Number of Pendent Vertices

Alraqad, Tariq; Milovanovic, Igor Z.; Saber, Hicham; Ali, Akbar; Mazorodze, Jaya Percival

doi:10.48550/arxiv.2211.05218

Cited by 4 publications

(4 citation statements)

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“…In the paper [12], graphs possessing the maximum and minimum values of the ABS index were characterized over the classes of graphs and (chemical) trees of a given order; such kind of extremal results regarding unicyclic graphs were found in [13], where also chemical applications of the ABS index were reported. The preprint [14] is concerned with the problems of determining graphs possessing the minimum ABS index among all trees of a fixed order and/or a given number of pendant vertices; see also [15] where one of these two problems is attacked independently.…”

Section: Introductionmentioning

confidence: 99%

On the Maximum Atom-Bond Sum-Connectivity Index of Graphs

Alraqad¹,

Saber²,

Ali³

2023

Preprint

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Let G be a graph with m edges, namely e 1 , • • • , e m . For i ∈ {1, • • • , m}, let d e i be the number of edges adjacent with e i . The sum of the numbers 1 − 2(d e i + 2) −1 over 1 ≤ i ≤ m is known as the atom-bond sum-connectivity (ABS) index of G. A pendant vertex of G is a vertex having degree 1. A subset S of the vertex set of G is said to be independent if the vertices of S are pairwise non-adjacent in G. The maximum number among cardinalities of all independent sets of G is known as the independence number of G. The least number of colors required to color the vertices of a graph, so that no two adjacent vertices have the same color, is termed as the chromatic number. This paper characterizes the graphs attaining the greatest values of the ABS index over the classes of graphs of a given order and with a fixed (i) chromatic number (ii) independence number (iii) number of pendant vertices.

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Section: Introductionmentioning

confidence: 99%

On the Maximum Atom-Bond Sum-Connectivity Index of Graphs

Alraqad¹,

Saber²,

Ali³

2023

Preprint

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show abstract

“…Article [14] not only gives a solution to an extremal problem involving the ABS index for unicyclic graphs, but it also reports chemical uses of the ABS index. The trees with the lowest ABS index were examined in [15,16] independently, with a specified number of vertices of degree 1 and a fixed order. Further existing results on the ABS index can be found in [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

General Atom-Bond Sum-Connectivity Index of Graphs

2023

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This paper is concerned with the general atom-bond sum-connectivity index ABSγ, which is a generalization of the recently proposed atom-bond sum-connectivity index, where γ is any real number. For a connected graph G with more than two vertices, the number ABSγ(G) is defined as the sum of (1−2(dx+dy)−1)γ over all edges xy of the graph G, where dx and dy represent the degrees of the vertices x and y of G, respectively. For −10≤γ≤10, the significance of ABSγ is examined on the data set of twenty-five benzenoid hydrocarbons for predicting their enthalpy of formation. It is found that the predictive ability of the index ABSγ for the selected property of the considered hydrocarbons is comparable to other existing general indices of this type. The effect of the addition of an edge between two non-adjacent vertices of a graph under ABSγ is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved.

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“…In [3], the graphs with the minimum and second-minimum ABS index are determined among all connected unicyclic graphs. In [5] and [17], the authors provided independently a partial solution to the problem of characterizing graphs possessing the minimum ABS index among all trees of a fixed order and number of pendant vertices.…”

Section: Introductionmentioning

confidence: 99%