Maximum values of atom–bond connectivity index in the class of tricyclic graphs

Ашрафи, Али Реза; Dehghan-Zadeh, T.; Habibi, Nader; John, Peter E.

doi:10.1007/s12190-015-0882-x

Cited by 8 publications

(7 citation statements)

References 12 publications

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“…Let us assume that n ≥ 5 and M ∼ = P n . Note that 1 ≤ d i ≤ 6 for all vertices v i of G. Hence the edges of G are of possible degree pairs: (6, 6), (6,5), (6,4), (6,3), (6, 2), (6, 1), (5,5), (5,4), (5,3), (5, 2), (5, 1), (4, 4), (4, 3), (4, 2), (4, 1), (3, 3), (3, 2), (3, 1), (2, 2), (2,1). The values of…”

Section: Comparison Between Ga Index and Abc Indexmentioning

confidence: 99%

“…Due to its physicochemical applicability, the ABC index has attracted significant attention from researchers in recent years and many mathematical properties of this index were reported. For instance, see the papers [3-5, 13, 14, 19, 22, 23], more precisely the recent ones [2,9,18,20] and references cited therein. A graph having maximum vertex degree at most 4 is known as a molecular graph.…”

Section: Introductionmentioning

confidence: 99%

“…It can be easily seen that no edge with degree pairs (2, 6) can be incident with any vertex of the clique K 4 . This implies, corresponding to every clique K 4 there exist at most 8 edges with degree pairs(2,5), but the size of K 4 is 6. Hence 8(0.1964) + 6(0.2727) 14 ≈ 0.2291 > 0.2130.…”

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confidence: 99%

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More on comparison between first geometric-arithmetic index and atom-bond connectivity index

Raza¹,

Bhatti²,

Ali³

2016

MMN

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The first geometric-arithmetic (GA) index and atom-bond connectivity (ABC) index are molecular structure descriptors which play a significant role in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies. Das and Trinajstić [Chem. Phys. Lett. 497 (2010) 149-151] showed that GA index is greater than ABC index for all those graphs (except K1,4 and T * , see Figure 1) in which the difference between maximum and minimum degree is less than or equal to 3. In this note, it is proved that GA index is greater than ABC index for line graphs of molecular graphs, for general graphs in which the difference between maximum and minimum degree is less than or equal to (2δ − 1) 2 (where δ is the minimum degree and δ ≥ 2) and for some families of trees. Therefore, a partial solution to an open problem proposed by Das and Trinajstić is given. , 2010 Mathematics Subject Classification. 05C07, 05C35, 92E10. Key words and phrases. first geometric-arithmetic index, atom-bond connectivity index, minimum and maximum degree of a graph, line graph.

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Section: Comparison Between Ga Index and Abc Indexmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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More on comparison between first geometric-arithmetic index and atom-bond connectivity index

Raza¹,

Bhatti²,

Ali³

2016

MMN

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“…In [6], the maximum and minimum ABC indices of all unicyclic graphs and unicyclic chemical graphs were obtained, and the corresponding extremal graphs were also characterized. In [7,8], the maximum values of the ABC indices in the class of all n-vertex bicyclic and tricyclic graphs were presented, respectively.…”

Section: Introductionmentioning

confidence: 99%

The Maximal ABC Index of the Corona of Two Graphs

Liu

Shao

2021

Mathematical Problems in Engineering

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Let G 1 ∘ G 2 be the corona of two graphs G 1 and G 2 which is the graph obtained by taking one copy of G 1 and V G 1 copies of G 2 and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2 . The atom-bond connectivity index (ABC index) of a graph G is defined as A B C G = ∑ u v ∈ E G d G u + d G v − 2 / d G u d G v , where E G is the edge set of G and d G u and d G v are degrees of vertices u and v , respectively. For the ABC indices of G 1 ∘ G 2 with G 1 and G 2 being connected graphs, we get the following results. (1) Let G 1 and G 2 be connected graphs. The ABC index of G 1 ∘ G 2 attains the maximum value if and only if both G 1 and G 2 are complete graphs. If the ABC index of G 1 ∘ G 2 attains the minimum value, then G 1 and G 2 must be trees. (2) Let T 1 and T 2 be trees. Then, the ABC index of T 1 ∘ T 2 attains the maximum value if and only if T 1 is a path and T 2 is a star.

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“…where d u is the degree of the vertex u ∈ V (G) and uv is edge connecting the vertices u and v. Details about this index can be found in the survey , recent papers (Ashrafi et al, 2015;Dimitrov, 2014;Lin et al, 2015;Palacios, 2014;Raza, Bhatti & Ali, 2015) and the related references cited therein.…”

Section: Introductionmentioning

confidence: 99%

A note on the augmented Zagreb index of cacti with fixed number of vertices and cycles

Ali,

Bhatti

2016

Preprint

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Maximum values of atom–bond connectivity index in the class of tricyclic graphs

Abstract: The atom-bond connectivity index of a graph G is defined aswhere d(u) is degree of the vertex u. In this paper, the first, second and third maximum values of the atom-bond connectivity index in the class of all n-vertex tricyclic graphs are presented.

Cited by 8 publications

References 12 publications

More on comparison between first geometric-arithmetic index and atom-bond connectivity index

More on comparison between first geometric-arithmetic index and atom-bond connectivity index

The Maximal ABC Index of the Corona of Two Graphs

A note on the augmented Zagreb index of cacti with fixed number of vertices and cycles

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