Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices

1993 ◽  
Vol 12 (1) ◽  
pp. 309-318 ◽  
Author(s):  
Ovidiu Ivanciuc ◽  
Teodor-Silviu Balaban ◽  
Alexandru T. Balaban
Keyword(s):  
Distance Matrix ◽  
Topological Indices ◽  
Local Vertex ◽  
Reciprocal Distance Matrix

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals The Wiener Index of Circulant Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Houqing Zhou
Keyword(s):  
Distributed Computing ◽  
Interconnection Networks ◽  
Distance Matrix ◽  
Wiener Index ◽  
Important Class ◽  
Circulant Graphs ◽  
Parallel And Distributed Computing ◽  
Harary Index ◽  
Largest Eigenvalues ◽  
Reciprocal Distance Matrix

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. In this paper, we discuss the relation of the Wiener index and the Harary index of circulant graphs and the largest eigenvalues of distance matrix and reciprocal distance matrix of circulants. We obtain the following consequence:W/λ=H/μ;2W/n=λ;2H/n=μ, whereW,Hdenote the Wiener index and the Harary index andλ,μdenote the largest eigenvalues of distance matrix and reciprocal distance matrix of circulant graphs, respectively. Moreover we also discuss the Wiener index of nonregular graphs with cut edges.

Maximum eigenvalue of the reciprocal distance matrix

2009 ◽  
Vol 47 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Kinkar Ch. Das
Keyword(s):  
Distance Matrix ◽  
Maximum Eigenvalue ◽  
Reciprocal Distance Matrix

Maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix

2008 ◽  
Vol 108 (5) ◽  
pp. 858-864 ◽  
Author(s):  
Bo Zhou ◽  
Nenad Trinajstić
Keyword(s):  
Distance Matrix ◽  
Reciprocal Distance Matrix

Some topological indices in fuzzy graphs

2020 ◽  
Vol 39 (5) ◽  
pp. 6033-6046
Author(s):  
Shriram Kalathian ◽  
Sujatha Ramalingam ◽  
Sundareswaran Raman ◽  
Narasimman Srinivasan
Keyword(s):  
Graph Theory ◽  
Classical Model ◽  
Distance Matrix ◽  
Wiener Index ◽  
Topological Indices ◽  
Fuzzy Graph ◽  
Structural Graph ◽  
Chemical Graph Theory ◽  
Fuzzy Graphs ◽  
Schultz Index

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

THE HARARY INDEX OF A GRAPH UNDER PERTURBATION

2010 ◽  
Vol 02 (02) ◽  
pp. 247-255 ◽  
Author(s):  
CHANG-XIANG HE ◽  
PING CHEN ◽  
BAO-FENG WU
Keyword(s):  
Distance Matrix ◽  
Harary Index ◽  
Reciprocal Distance Matrix

The Harary index is defined as the half-sum of the elements in the reciprocal distance matrix. In this paper, we investigate how the Harary index behaves when the graph is perturbed by grafting or moving edges.

Topologische Indices alternierender polycyclischer aromatischer Kohlenwasserstoffe und ihre Korrelationen mit elektronischen Eigenschaften / Topological Indices of Alternant Polycyclic Aromatic Hydrocarbons and their Correlations with Electronic Properties

1981 ◽  
Vol 36 (11) ◽  
pp. 1217-1221
Author(s):  
K.-D. Gundermann ◽  
C. Lohberger ◽  
M. Zander
Keyword(s):  
Polycyclic Aromatic Hydrocarbons ◽  
Electronic Properties ◽  
Aromatic Hydrocarbons ◽  
Topological Index ◽  
Distance Matrix ◽  
Topological Indices ◽  
Matrix Elements ◽  
Wiener Number ◽  
Polycyclic Aromatic ◽  
Aromatic Systems

The half-sum of the distance matrix elements derived from the characteristic graphs, i.e. the Wiener number of these graphs is proposed as a new topological index for alternant polycyclic aromatic hydrocarbons. It is shown by regression analysis that correlations between topological indices and electronic properties of alternant aromatic systems do only exist for those indices and properties which depend to the same degree from the size of the systems and for which the corresponding relation applies to the topology.

scholarly journals Molecular van der Waals Space and Topological Indices from the Distance Matrix

Molecules ◽  
2004 ◽  
Vol 9 (12) ◽  
pp. 1053-1078 ◽  
Author(s):  
Dan Ciubotariu ◽  
Mihai Medeleanu ◽  
Vicentiu Vlaia ◽  
Tudor Olariu ◽  
Ciprian Ciubotariu ◽  
...  
Keyword(s):  
Van Der Waals ◽  
Distance Matrix ◽  
Topological Indices

Bounds and extremal graphs for Harary energy

2021 ◽  
pp. 2150149
Author(s):  
A. Alhevaz ◽  
M. Baghipur ◽  
H. A. Ganie ◽  
K. C. Das
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Energy Formula ◽  
Matrix Formula ◽  
The Absolute ◽  
Graph Parameters ◽  
Reciprocal Distance Matrix

Let [Formula: see text] be a connected graph of order [Formula: see text] and let [Formula: see text] be the reciprocal distance matrix (also called Harary matrix) of the graph [Formula: see text]. Let [Formula: see text] be the eigenvalues of the reciprocal distance matrix [Formula: see text] of the connected graph [Formula: see text] called the reciprocal distance eigenvalues of [Formula: see text]. The Harary energy [Formula: see text] of a connected graph [Formula: see text] is defined as sum of the absolute values of the reciprocal distance eigenvalues of [Formula: see text], that is, [Formula: see text] In this paper, we establish some new lower and upper bounds for [Formula: see text] in terms of different graph parameters associated with the structure of the graph [Formula: see text]. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of [Formula: see text] largest adjacency eigenvalues of a connected graph.

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