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Analytical and Computational Properties of Topological Indices II

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 March 2022) | Viewed by 5434

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Dear Colleagues,

Although Topological Indices have played an important role in Mathematical Chemistry since the seminal work of Wiener in 1947, in recent years, this role has significantly increased. On the one side, molecular descriptors constitute an aid tool in Chemistry, especially in QSPR/QSAR investigations. On the other side, they have become an important part of some areas of Mathematics, as Graph Theory; this interest has been recognized in the 2020-version of the Mathematical Subject Classification by including two new areas: 05C09 - Graphical indices (Wiener index, Zagreb index, Randić index, etc.), and 05C92 - Chemical graph theory.

The aim of this Special Issue is to attract leading researchers in this area in order to include new results on these topics, both from a theoretical and an applied point of view.

Prof. Dr. Jose M. Rodriguez
Prof. Dr. Eva Tourís
Guest Editors

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Keywords

topological indices
graphical indices
chemical graph theory
mathematical chemistry
topological descriptors
molecular descriptors
graph optimization problems
polynomials on topological indices

Published Papers (3 papers)

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Research

18 pages, 941 KiB

Open AccessArticle

New Versions of Locating Indices and Their Significance in Predicting the Physicochemical Properties of Benzenoid Hydrocarbons

by Suha Wazzan and Anwar Saleh

Symmetry 2022, 14(5), 1022; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14051022 - 17 May 2022

Cited by 6 | Viewed by 1918

Abstract

In this paper, we introduce some new versions based on the locating vectors named locating indices. In particular, Hyper locating indices, Randić locating index, and Sambor locating index. The exact formulae for these indices of some well-known families of graphs and for the Helm graph are derived. Moreover, we determine the importance of these locating indices for 11 benzenoid hydrocarbons. Furthermore, we show that these new versions of locating indices have a reasonable correlation using linear regression with physicochemical characteristics such as molar entropy, acentric factor, boiling point, complexity, octanol–water partition coefficient, and Kovats retention index. The cases in which good correlations were obtained suggested the validity of the calculated topological indices to be further used to predict the physicochemical properties of much more complicated chemical compounds. Full article

(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices II)

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13 pages, 325 KiB

Open AccessArticle

Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

by Juan Monsalve and Juan Rada

Symmetry 2021, 13(10), 1903; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101903 - 09 Oct 2021

Cited by 7 | Viewed by 1192

Abstract

A vertex-degree-based (VDB, for short) topological index

φ

induced by the numbers

\{φ_{i j}\}

was recently defined for a digraph D, as

φ (D) = \frac{1}{2} \sum_{u v} φ_{d_{u}^{+} d_{v}^{-}},

where [...] Read more.

A vertex-degree-based (VDB, for short) topological index

φ

induced by the numbers

\{φ_{i j}\}

was recently defined for a digraph D, as

φ (D) = \frac{1}{2} \sum_{u v} φ_{d_{u}^{+} d_{v}^{-}},

where

d_{u}^{+}

denotes the out-degree of the vertex

u,

d_{v}^{-}

denotes the in-degree of the vertex

v,

and the sum runs over the set of arcs

u v

of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over

D_{n}

, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over

OT (n)

and

O (G)

, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively. Full article

(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices II)

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13 pages, 326 KiB

Open AccessArticle

On a Conjecture about the Sombor Index of Graphs

by Kinkar Chandra Das, Ali Ghalavand and Ali Reza Ashrafi

Symmetry 2021, 13(10), 1830; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101830 - 01 Oct 2021

Cited by 14 | Viewed by 1595

Abstract

Let G be a graph with vertex set

V (G)

and edge set

E (G)

. A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. [...] Read more.

Let G be a graph with vertex set

V (G)

and edge set

E (G)

. A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as

S O (G) = \sum_{u v \in E (G)} \sqrt{d_{G} {(u)}^{2} + d_{G} {(v)}^{2}}

and

S O_{r e d} (G) = \sum_{u v \in E (G)} \sqrt{{(d_{G} (u) - 1)}^{2} + {(d_{G} (v) - 1)}^{2}}

, respectively, where

d_{G} (v)

is the degree of the vertex v in G. We denote by

H_{n, ν}

the graph constructed from the star

S_{n}

by adding

ν

edge(s),

0 \leq ν \leq n - 2

, between a fixed pendent vertex and

ν

other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph

H_{n, ν}

has the maximum Sombor index among all connected

ν

-cyclic graphs of order n, where

0 \leq ν \leq n - 2

. In some earlier works, the validity of this conjecture was proved for

ν \leq 5

. In this paper, we confirm that this conjecture is true, when

ν = 6

. The Sombor index in the case that the number of pendent vertices is less than or equal to

n - ν - 2

is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained. Full article

(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices II)

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