Maximal bond incident degree index for trees and unicyclic graphs with fixed diameter

Zhenhua Su; Zhenhua Su

doi:10.3934/math.2026204

AIMS Mathematics

2026, Volume 11, Issue 2: 4985-5005. doi: 10.3934/math.2026204

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Maximal bond incident degree index for trees and unicyclic graphs with fixed diameter

Zhenhua Su ^{1,2
,
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1.
School of Mathematics and Computational Sciences, Huaihua University, Huaihua, Hunan, China
2.
Key Laboratory of Intelligent Control Technology for Wuling-Mountain Ecological Agriculture in Hunan Province, Huaihua, Hunan, China

Received: 22 December 2025 Revised: 31 January 2026 Accepted: 11 February 2026 Published: 27 February 2026
MSC : 05C05, 05C09, 05C92

The bond incident degree index of $ G $ is defined as
$ BID(G) = \sum\limits_{u_1u_2\in E(G)} f(d(u_1),d(u_2)), $
where $ f(y, x) = f(x, y) $ is a real-valued function. In this paper, using graph transformation methods, we respectively established the maximum bond incident degree indices of trees and unicyclic graphs with a fixed diameter. As an application of the sufficient conditions, we verified that six bond incident degree indices satisfy such conditions, among which are the newly introduced Euler Sombor index and the computationally complex general Sombor index.
- bond incident degree index,
- trees,
- unicyclic graphs,
- diameter
Citation: Zhenhua Su. Maximal bond incident degree index for trees and unicyclic graphs with fixed diameter[J]. AIMS Mathematics, 2026, 11(2): 4985-5005. doi: 10.3934/math.2026204

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Abstract

The bond incident degree index of $ G $ is defined as

$ BID(G) = \sum\limits_{u_1u_2\in E(G)} f(d(u_1),d(u_2)), $

where $ f(y, x) = f(x, y) $ is a real-valued function. In this paper, using graph transformation methods, we respectively established the maximum bond incident degree indices of trees and unicyclic graphs with a fixed diameter. As an application of the sufficient conditions, we verified that six bond incident degree indices satisfy such conditions, among which are the newly introduced Euler Sombor index and the computationally complex general Sombor index.

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