INVERSE DEGREE INVARIANT OF GRAPHS

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of n -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.

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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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Leaves and inverse degree of a graph

2010 3rd International Conference on Biomedical Engineering and Informatics ◽

10.1109/bmei.2010.5639359 ◽

2010 ◽

Author(s):

Bing Yao ◽

Xiang-en Chen ◽

Ming Yao ◽

Jiajuan Zhang ◽

Jingxia Guo

Keyword(s):

Inverse Degree

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Inverse degree, Randic index and harmonic index of graphs

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm1702304d ◽

2017 ◽

Vol 11 (2) ◽

pp. 304-313 ◽

Cited By ~ 6

Author(s):

Kinkar Das ◽

Selvaraj Balachandran ◽

Ivan Gutman

Keyword(s):

Randić Index ◽

Harmonic Index ◽

Randic Index ◽

Vertex Set ◽

Inverse Degree

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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Inequalities on the inverse degree index

Journal of Mathematical Chemistry ◽

10.1007/s10910-019-01022-3 ◽

2019 ◽

Vol 57 (5) ◽

pp. 1524-1542 ◽

Cited By ~ 3

Author(s):

José M. Rodríguez ◽

José L. Sánchez ◽

José M. Sigarreta

Keyword(s):

Inverse Degree

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f-Polynomial on Some Graph Operations

Mathematics ◽

10.3390/math7111074 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1074 ◽

Cited By ~ 1

Author(s):

Walter Carballosa ◽

José Manuel Rodríguez ◽

José María Sigarreta ◽

Nodari Vakhania

Keyword(s):

Large Family ◽

Topological Indices ◽

Graph Operations ◽

Inverse Degree ◽

Corona Product ◽

Mathematical Relations

Given any function f : Z + → R + , let us define the f-index I f ( G ) = ∑ u ∈ V ( G ) f ( d u ) and the f-polynomial P f ( G , x ) = ∑ u ∈ V ( G ) x 1 / f ( d u ) − 1 , for x > 0 . In addition, we define P f ( G , 0 ) = lim x → 0 + P f ( G , x ) . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others.

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On the Inverse Degree Polynomial

Symmetry ◽

10.3390/sym11121490 ◽

2019 ◽

Vol 11 (12) ◽

pp. 1490

Author(s):

Paul Bosch ◽

José Manuel Rodríguez ◽

Omar Rosario ◽

José María Sigarreta

Keyword(s):

Symmetry Property ◽

Degree Polynomial ◽

Cyclomatic Number ◽

Inverse Degree ◽

Mathematical Relations

Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials.

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