The Maximal ABC Index of the Corona of Two Graphs-Reference-Cited by-同舟云学术

The Maximal ABC Index of the Corona of Two Graphs

Published:2021-04-01 Issue: Volume:2021 Page:1-9
ISSN:1563-5147
Container-title:Mathematical Problems in Engineering
language:en
Short-container-title:Mathematical Problems in Engineering

Author:

Liu Haiqin¹²^ORCID,Shao Yanling³^ORCID

Affiliation:

1. College of Big Data, North University of China, Taiyuan 030051, Shanxi, China

2. Basic Department, Shanxi Agricultural University, Taigu 030801, Shanxi, China

3. School of Science, North University of China, Taiyuan 030051, Shanxi, China

Abstract

Let

G_{1} \circ G_{2}

be the corona of two graphs

G_{1}

and

G_{2}

which is the graph obtained by taking one copy of

G_{1}

and

|V (G_{1})|

copies of

G_{2}

and then joining the

i

th vertex of

G_{1}

to every vertex in the

i th

copy of

G_{2}

. The atom-bond connectivity index (ABC index) of a graph

G

is defined as

A B C (G) = \sum_{u v \in E (G)} \sqrt{(d_{G} (u) + d_{G} (v) - 2 / d_{G} (u) d_{G} (v))}

, where

E (G)

is the edge set of

G

and

d_{G} (u)

and

d_{G} (v)

are degrees of vertices

u

and

v

, respectively. For the ABC indices of

G_{1} \circ G_{2}

with

G_{1}

and

G_{2}

being connected graphs, we get the following results. (1) Let

G_{1}

and

G_{2}

be connected graphs. The ABC index of

G_{1} \circ G_{2}

attains the maximum value if and only if both

G_{1}

and

G_{2}

are complete graphs. If the ABC index of

G_{1} \circ G_{2}

attains the minimum value, then

G_{1}

and

G_{2}

must be trees. (2) Let

T_{1}

and

T_{2}

be trees. Then, the ABC index of

T_{1} \circ T_{2}

attains the maximum value if and only if

T_{1}

is a path and

T_{2}

is a star.

Funder

Shanxi Province Science Foundation for Youths

Publisher

Hindawi Limited

Subject

General Engineering,General Mathematics

Link

http://downloads.hindawi.com/journals/mpe/2021/6699529.pdf

Reference19 articles.

1. On the corona of two graphs

2. On atom-bond connectivity index

3. Extreme atom-bond connectivity index of graphs;J. Chen;MATCH Communications in Mathematical and in Computer Chemistry,2011

4. The smallest ABC index of trees with n pendent vertices;Y. Gao;MATCH Communications in Mathematical and in Computer Chemistry,2016

5. Atom–bond connectivity index of trees