Inverse degree, Randic index and harmonic index of graphs

Let G be a graph with vertex set V and edge set E. Let di be the degree of the vertex vi of G. The inverse degree, Randic index, and harmonic index of G are defined as ID = ?vi?V 1/di, R = ? vivj?E 1/?di dj , and H = ? vivj?E 2=(di + dj), respectively. We obtain relations between ID and R as well as between ID and H. Moreover, we prove that in the case of trees, ID > R and ID > H.

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Some tight bounds for the harmonic index and the variation of the Randić index of graphs

Discrete Mathematics ◽

10.1016/j.disc.2019.03.022 ◽

2019 ◽

Vol 342 (7) ◽

pp. 2060-2065

Author(s):

Hanyuan Deng ◽

Selvaraj Balachandran ◽

Suresh Elumalai

Keyword(s):

Randić Index ◽

Harmonic Index ◽

Randic Index

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Some Bounds on Zeroth-Order General Randić Index

Mathematics ◽

10.3390/math8010098 ◽

2020 ◽

Vol 8 (1) ◽

pp. 98 ◽

Cited By ~ 2

Author(s):

Muhammad Kamran Jamil ◽

Ioan Tomescu ◽

Muhammad Imran ◽

Aisha Javed

Keyword(s):

Clique Number ◽

Upper Bounds ◽

Randić Index ◽

Extremal Graphs ◽

Lower And Upper Bounds ◽

Zeroth Order ◽

Randic Index ◽

General Randić Index ◽

Inverse Degree ◽

General Randic Index

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = ∑ u ∈ V ( G ) d ( u ) − 1 where d ( u ) is the number of vertices adjacent to the vertex u in G. By replacing − 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R γ ( G ) = ∑ u ∈ V ( G ) d ( u ) γ , where γ ∈ R − { 0 } . Xu et al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for γ < 0 . The corresponding extremal graphs have also been identified.

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Functions on adjacent vertex degrees of trees with given degree sequence

Open Mathematics ◽

10.2478/s11533-014-0439-5 ◽

2014 ◽

Vol 12 (11) ◽

Cited By ~ 1

Author(s):

Hua Wang

Keyword(s):

Symmetric Function ◽

Degree Sequence ◽

Connectivity Index ◽

Adjacent Vertex ◽

Randić Index ◽

Graph Invariant ◽

Harmonic Index ◽

Randic Index ◽

Special Cases ◽

Vertex Degrees

AbstractIn this note we consider a discrete symmetric function f(x, y) where $$f(x,a) + f(y,b) \geqslant f(y,a) + f(x,b) for any x \geqslant y and a \geqslant b,$$ associated with the degrees of adjacent vertices in a tree. The extremal trees with respect to the corresponding graph invariant, defined as $$\sum\limits_{uv \in E(T)} {f(deg(u),deg(v))} ,$$ are characterized by the “greedy tree” and “alternating greedy tree”. This is achieved through simple generalizations of previously used ideas on similar questions. As special cases, the already known extremal structures of the Randic index follow as corollaries. The extremal structures for the relatively new sum-connectivity index and harmonic index also follow immediately, some of these extremal structures have not been identified in previous studies.

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On the Randić Index of Corona Product Graphs

ISRN Discrete Mathematics ◽

10.5402/2011/262183 ◽

2011 ◽

Vol 2011 ◽

pp. 1-7

Author(s):

Ismael G. Yero ◽

Juan A. Rodríguez-Velázquez

Keyword(s):

Degree Sequence ◽

Upper Bounds ◽

Vertex Degree ◽

Regular Graphs ◽

Randić Index ◽

Lower And Upper Bounds ◽

Randic Index ◽

Product Graphs ◽

Vertex Set ◽

Corona Product

Let G be a graph with vertex set V=(v1,v2,…,vn). Let δ(vi) be the degree of the vertex vi∈V. If the vertices vi1,vi2,…,vih+1 form a path of length h≥1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/δ(vi1)δ(vi2)⋯δ(vih+1) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

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Relationship Between Randić Index, Sum-Connectivity Index, Harmonic Index and $$\pi $$π-Electron Energy for Benzenoid Hydrocarbons

National Academy Science Letters ◽

10.1007/s40009-019-0782-y ◽

2019 ◽

Vol 42 (6) ◽

pp. 519-524 ◽

Cited By ~ 1

Author(s):

H. S. Ramane ◽

V. B. Joshi ◽

R. B. Jummannaver ◽

S. D. Shindhe

Keyword(s):

Electron Energy ◽

Connectivity Index ◽

Randić Index ◽

Benzenoid Hydrocarbons ◽

Harmonic Index ◽

Randic Index

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On Multiplicative Sum connectivity index, Multiplicative Randic index and Multiplicative Harmonic index of some Nanostar Dendrimers

International Journal of Engineering Science Advanced Computing and Bio-Technology ◽

10.26674/ijesacbt/2018/49410 ◽

2018 ◽

Vol 9 (2) ◽

Cited By ~ 2

Author(s):

M. Bhanumathi ◽

K. Easu Julia Rani

Keyword(s):

Connectivity Index ◽

Randić Index ◽

Harmonic Index ◽

Randic Index

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M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide

Journal of Chemistry ◽

10.1155/2021/6679819 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Faryal Chaudhry ◽

Iqra Shoukat ◽

Deeba Afzal ◽

Choonkil Park ◽

Murat Cancan ◽

...

Keyword(s):

Topological Indices ◽

Randić Index ◽

Zagreb Index ◽

Harmonic Index ◽

Randic Index ◽

Zagreb Indices ◽

Symmetric Division ◽

Augmented Zagreb Index ◽

Physical And Chemical ◽

Modified Zagreb Index

Topological indices are numerical parameters used to study the physical and chemical residences of compounds. Degree-based topological indices have been studied extensively and can be correlated with many properties of the understudy compounds. In the factors of degree-based topological indices, M-polynomial played an important role. In this paper, we derived closed formulas for some well-known degree-based topological indices like first and second Zagreb indices, the modified Zagreb index, the symmetric division index, the harmonic index, the Randić index and inverse Randić index, and the augmented Zagreb index using calculus.

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More on inverse degree and topological indices of graphs

Filomat ◽

10.2298/fil1801165e ◽

2018 ◽

Vol 32 (1) ◽

pp. 165-178

Author(s):

Suresh Elumalai ◽

Sunilkumar Hosamani ◽

Toufik Mansour ◽

Mohammad Rostami

Keyword(s):

Upper Bounds ◽

Topological Indices ◽

Computational Results ◽

Randić Index ◽

Lower And Upper Bounds ◽

Randic Index ◽

The Third ◽

Inverse Degree ◽

Vertex Degrees ◽

Chemical Trees

The inverse degree of a graph G with no isolated vertices is defined as the sum of reciprocal of vertex degrees of the graph G. In this paper, we obtain several lower and upper bounds on inverse degree ID(G). Moreover, using computational results, we prove our upper bound is strong and has the smallest deviation from the inverse degree ID(G). Next, we compare inverse degree ID(G) with topological indices (Randic index R(G), geometric-arithmetic index GA(G)) for chemical trees and also we determine the n-vertex chemical trees with the minimum, the second and the third minimum, as well as the second and the third maximum of ID - R. In addition, we correct the second and third minimum Randic index chemical trees in [16].

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