SOMBOR INDEX OF LINE AND TOTAL GRAPHS AND PERICONDENSED BENZENOID HYDROCARBONS
Yue Li ^{1 }, Qingcuo Ren ^{1}, Jinxia Liang^{ 1}, Chengxu Yang ^{2}, Qinghe Tong^{ 1}
^{1}School of Mathematics and
Statistics, Qinghai Normal University, Xining, Qinghai 810008, China
^{2}^{ }School of Computer, Qinghai Normal
University, Xining, Qinghai 810008, China

ABSTRACT 

Gutman
proposed a new alternative interpretation of vertexdegreebased topological
index, called Sombor index. It is defined via the
term . In this paper, we
determine the explicit expressions of Sombor index for line and total graphs
and several pericondensed benzenoid hydrocarbons. 

Received 20 July 2022 Accepted 22 August 2022 Published 07 September 2022 Corresponding Author Jinxia Liang, ljxqhsd@aliyun.com DOI 10.29121/granthaalayah.v10.i8.2022.4730 Funding: The third
author was supported by the National Science Foundation of China (Nos.
11601254, 11551001, 11161037, 61763041, 11661068, and 11461054) and the
Qinghai Key Laboratory of Internet of Things Project (2017ZJY21). Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CCBY, authors retain the copyright, allowing anyone to download,
reuse, reprint, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. 

Keywords: Sombor Index,
Chemical Indicator, Pericondensed Benzenoid, Hydrocarbons 
1. INTRODUCTION
In the mathematical and chemical literature, several dozens of vertexdegreebased graph invariants have been introduced and extensively studied in Pal et al. (2019), Todeschini and Consonni (2009). For a graph , let , and and denote the size, the minimum degree and the maximum degree and the degree of the vertex , respectively. The line graph is the graph whose vertex set is the edges of, two vertices and of being adjacent if and only if corresponding edges in are adjacent. The total graph of a graph is the graph whose vertex set is with two vertices of being adjacent if and only if the corresponding elements of are either adjacent or incident.
Recently, Gutman (2021) introduced a new index defined as
Equation 1
called Sombor index.
The distance Gutman (2021) between the dpoint and the origin of the coordinate system is the degreeradius (or dradius) of the edge , denoted by Based on elementary geometry
(Using Euclidean metrics), we have
In Gutman (2021), Gutman presented a novel approach to the vertexdegreebased topological indices of (molecular) graphs. The upper and lower bounds of Sombor index for general trees and graphs are given, and some basic properties of the Sombor index are established. Cruza et al. (2021) characterized the graphs extremal with respect to the Sombor index over the following sets: (connected) chemical graphs, chemical trees, and hexagonal systems. Das and Gutman Das and Gutman (2022) presented bounds on SO index of trees in terms of order, independence number, and number of pendent vertices, and characterize the extremal cases. The mathematical relations between the Sombor index and some other wellknown degreebased descriptors was investigated in Wang et al. (2022).
In 2015, Su and Xu (2015) studied the general sumconnectivity index and coindex of line graph of subdivision graphs. In 2021, Demirci et al. (2021) obtained the explicit expressions for the Omega index of line and total graphs. In Section 2, the Sombor index of and are determined, respectively.
Klavˇzar et al. (1997)determined the explicit expressions of Wiener index for
several pericondensed benzenoid hydrocarbons.
We also determine the explicit expressions of
Sombor index for several pericondensed benzenoid hydrocarbons in Section 3.
2. RESULTS FOR LINE AND TOTAL GRAPHS
From the definitions, the following observation is immediate.
Observation 1. Let be a graph, , and . Then
and .
Theorem 2.1. Let be a connected graph of order n, with maximum degree and minimum degree . Then
with equality if and only if is a regular graph.
Proof. From the definition of Sombor index, we have
Since , it follows that
with equality if and only if .
For any vertex , let denote the set of vertices associated with v. Since and , it follows that
Similarly, to Theorem 2.1, we can give a lower bound of without its proof.
Observation 2. Let be a graph, , . Then and .
Theorem 2.2. Let be a connected graph of order n with m edges such that its maximum and minimum degrees are and , respectively. Then
with equality if and only if is a regular graph.
Proof. Let
From the definition of Sombor index, we have
Since , it follows that
For any vertex , since and
, it follows that
and hence
with equality if and only if is a regular graph.
3. RESULTS FOR PERICONDENSED BENZENOID HYDROCARBONS
In this section, we determine the explicit exact values for Sombor index of several pericondensed benzenoid hydrocarbons.
3.1. PARALLELOGRAM BENZENOID SYSTEM
For and ,
let be the parallelogram benzenoid system. The
definition of should be clear from the example shown in Figure 1, Klavˇzar et al. (1997).
Figure 1
Figure 1 Parallelogram Benzenoid System 
Theorem 3.1. Let be two integers with and . Let be the parallelogram benzenoid system. Then
Proof. For , we have . From the definition of Sombor index, we have
3.2. TRAPEZIUM BENZENOID SYSTEM
For and , let be the trapezium benzenoid system. The definition
of should be clear from the example shown in Figure 2, Klavˇzar (1997).
Figure 2
Figure 2 Trapezium Benzenoid System 
Theorem 3.2. Let be two integers with and . Let be the trapezium benzenoid system. Then
Proof. For , we have , and hence
3.3. PARALLELOGRAMLIKE BENZENOID SYSTEMS
For and , let be the parallelogramlike benzenoid system of
type 1. The definition of should be clear from the example shown in Figure 3, Klavˇzar (1997).
Figure 3
Figure 3 ParallelogramLike Benzenoid System 
Theorem 3.3. Let be two integers with and . Let be the parallelogramlike benzenoid system of type 1. Then
Proof. From the definition of Sombor index, we have
For and , let be the parallelogramlike benzenoid system of type 2. The definition of should be clear from the example shown in Figure 3, Klavˇzar (1997).
Theorem 3.4. Let be two integers with and . Let be the parallelogramlike benzenoid system of type 2. Then
Figure
4
Figure 4 ParallelogramLike Benzenoid System of 
Proof. From the definition of Sombor index, we have
For and , let be the parallelogramlike benzenoid system of type 3. The definition of should be clear from the example shown in Figure 5, Klavˇzar (1997).
Theorem 3.5. Let be two integers with and . Let be the parallelogramlike benzenoid system of type 3. Then
Figure
5
Figure 5 ParallelogramLike Benzenoid System 
Proof. From the definition of Sombor index, we have
3.4. BITRAPEZIUM
BENZENOID SYSTEM
For ,，, and , let be the bitrapezium benzenoid system. The definition of should be clear from the example shown in Figure 6, Klavˇzar (1997).
Theorem 3.6. Let be three integers with and , and . Let be a bitrapezium benzenoid system. Then
Figure 6
Figure 6 Bitrapezium Benzenoid System 
Proof. From the definition of Sombor index, we have
3.5. GENERAL BENZENOID SYSTEM
For ,,, and , let be the bitrapezium benzenoid system. The definition of should be clear from the example shown in Figure 7, Klavˇzar (1997).
Figure 7
Figure 7 Bitrapezium Benzenoid System 
Theorem 3.7. Let be five integers with ,, and . Let be a general benzenoid system. Then
Proof. From the definition of Sombor index, we have
3.6. LPOLYGONAL CHAIN
Let A polygonal chain of cycles (polygons) is obtained from a sequence of cycles, , by adding a bridge to each pair of consecutive cycles. If all such cycles are cycles, then this polygonal chain is called an polygonal chain of length and denoted by. The cycle will be called the th polygon of . Note that, there are many ways to add a bridge between two consecutive cycles. So may not be unique when . But is unique when . The definition of should be clear from the shown in Figure 8, Wei and Shiu (2019).
Figure 8
Figure 8 LPolygonal Chain 
Theorem 3.8. Let be an polygonal chain (of length ). Then
Proof. From the definition of Sombor index, we have
3.7. TITANIA NANOTUBES
Titania nanotubes are comprehensively studied in materials science. The sheets with a thickness of a few atomic layers were found to be remarkably stable. Let be the rows and columns of the titanium nanotubes. The definition of should be clear from the shown in Figure 9. Imran et al. (2021).
Theorem 3.9. Let denote the graph of titanium nanotubes with m rows and n columns. Then
Figure 9
Figure 9 Titanium Nanotubes 
Proof. From the definition of Sombor index, we have
.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
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