Determining Maximum Values of Sombor-Index-Like Graph Invariants for Trees and Connected Graphs

Conceitos essenciais

The paper establishes the maximum values of the Sombor-index-like graph invariants SO5 and SO6 within the set of molecular trees with a given number of vertices, and determines the maximum value of SO5 within the set of graphs obtained by applying the join operation to two specific graphs of a given order.

Resumo

The paper focuses on analyzing the maximum values of the Sombor-index-like graph invariants SO5 and SO6 in different classes of graphs.
Key highlights:

The authors establish the maximum values of SO5 and SO6 within the set of molecular trees (trees with maximum degree ≤ 4) with a given number of vertices.
They determine the maximum value of SO5 within the set of graphs obtained by applying the join operation to two specific graphs of a given order.
The proofs rely on connections between number theory, polynomial theory, and multivariable function analysis, covering numerous distinct cases.
Fully classifying the set of connected graphs with the maximum SO5 is identified as a more demanding task, involving the examination of complex multivariable functions and a larger number of cases.
The analysis of univariate functions is reduced to the study of fifth-degree polynomials, which generally do not have solutions in all cases.

Estatísticas

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Citações

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Principais Insights Extraídos De

Maximum values of the Sombor-index-like graph invariants of trees and connected graphs

by Mila... às arxiv.org 04-09-2024

https://arxiv.org/pdf/2305.02925.pdf

Perguntas Mais Profundas

How can the methods and insights from this paper be extended to determine the maximum values of other Sombor-index-like graph invariants, such as SO1, SO2, SO3, and SO4, within the class of connected graphs

The methods and insights from this paper can be extended to determine the maximum values of other Sombor-index-like graph invariants within the class of connected graphs by following a similar approach. For each specific invariant, such as SO1, SO2, SO3, and SO4, the degree-based formulas can be adapted to calculate the maximum values within the set of connected graphs. This would involve defining the new invariants based on vertex degrees, establishing the constraints and conditions for the graphs under consideration, and then deriving the expressions for the maximum values. By applying the same principles of geometric reasoning and mathematical analysis used in this paper, researchers can explore the extremal properties of these additional graph invariants and identify the graphs that achieve the maximum values.

What are the potential applications of these Sombor-index-like graph invariants in areas beyond mathematics and chemistry, such as network analysis or data science

The Sombor-index-like graph invariants introduced in this paper have potential applications beyond mathematics and chemistry, particularly in areas such as network analysis and data science. These invariants provide valuable insights into the structural properties of graphs and can be used to characterize and compare different types of networks. In network analysis, these invariants can help in identifying key nodes or structures within a network, assessing network robustness and connectivity, and detecting anomalies or patterns in network data. In data science, these graph invariants can be utilized for feature extraction, dimensionality reduction, and clustering of complex data sets represented as graphs. By leveraging the unique properties of Sombor-index-like graph invariants, researchers can enhance their understanding of network dynamics, optimize network design, and improve data analysis techniques in various domains.

Can the techniques used in this paper be adapted to study the extremal properties of other classes of graph invariants that are not necessarily vertex-degree-based

The techniques used in this paper can be adapted to study the extremal properties of other classes of graph invariants that are not necessarily vertex-degree-based. By applying the principles of mathematical analysis, optimization, and constraint satisfaction to different types of graph invariants, researchers can explore the maximum and minimum values of these invariants within specific classes of graphs. This may involve formulating new mathematical expressions for the graph invariants, defining the constraints and conditions for the graphs under consideration, and developing algorithms to identify extremal graphs. By extending the methodology used in this paper to other classes of graph invariants, researchers can uncover valuable insights into the structural properties of graphs, identify optimal graph configurations, and advance the field of graph theory in diverse applications.