The lower bound of revised edge-Szeged index of unicyclic graphs with given diameter

doi:10.3868/s140-DDD-023-0020-x

Front. Math. China

2023, Vol. 18

Issue (4) : 251-275 https://doi.org/10.3868/s140-DDD-023-0020-x

RESEARCH　ARTICLE

The lower bound of revised edge-Szeged index of unicyclic graphs with given diameter

Min WANG, Mengmeng LIU(

)

Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730000, China

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Abstract

Given a connected graph $G$ , the revised edge-revised Szeged index is defined as $S z e ∗ (G) = ∑ e = u v ∈ E G (m u (e) + m 0 (e) 2) (m v (e) + m 0 (e) 2)$ , where $m u (e)$ , $m v (e)$ and $m 0 (e)$ are the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ , the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$ and the number of edges of $G$ at the same distance to $u$ and $v$ , respectively. In this paper, by transformation and calculation, the lower bound of revised edge-Szeged index of unicyclic graphs with given diameter is obtained, and the extremal graph is depicted.

Keywords Wiener index revised edge Szeged index unicyclic graph extremal graph

Corresponding Author(s): Mengmeng LIU

Online First Date: 07 December 2023 Issue Date: 12 December 2023

Cite this article:

Min WANG,Mengmeng LIU. The lower bound of revised edge-Szeged index of unicyclic graphs with given diameter[J]. Front. Math. China, 2023, 18(4): 251-275.

URL:

https://academic.hep.com.cn/fmc/EN/10.3868/s140-DDD-023-0020-x
https://academic.hep.com.cn/fmc/EN/Y2023/V18/I4/251

Fig.1

ω 1 ω 2

G

is the cut edge, and

ω 1 ω 2

G ′

is the pendant edge.

Fig.2

C 4

and

P d + 1

have a common vertex, that is

u ? d 2 ?

, and the

u ? d 2 ?

has

m ? d ? 4

pendant edges.

Fig.3 The common vertices between

C 4

and

P d + 1

G 2

are

u ? d 2 ?

and

u ? d 2 ? + 1

. The common vertices between

C 4

and

P d + 1

G 3

are

u ? d 2 ?, u ? d 2 ? + 1

Fig.4 The common vertices between

C 4

and

P d + 1

G 4

are

u ? d 2 ? + 1

u ? d 2 ? + 2

and

u ? d 2 ? + 3

. The common vertices between

C 4

and

P d + 1

G 5

are

u d ? 2

u d ? 1

and

u d

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