Extremal Values of the Atom-Bond Connectivity Index for Trees with Given Roman Domination Numbers
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915024754180096 |
|---|---|
| author | Ali, Waqar Husin, Mohamad Nazri Bin Nadeem, Muhammad Faisal |
| author_facet | Ali, Waqar Husin, Mohamad Nazri Bin Nadeem, Muhammad Faisal |
| contents | Consider that $\mathbb{G}=(\mathbb{X}, \mathbb{Y})$ is a simple, connected graph with $\mathbb{X}$ as the vertex set and $\mathbb{Y}$ as the edge set. The atom-bond connectivity ($ABC$) index is a novel topological index that Estrada introduced in Estrada et al. (1998). It is defined as $$ A B C(\mathbb{G})=\sum_{xy \in Y(\mathbb{G})} \sqrt{\frac{ζ_x+ζ_y-2}{ζ_x ζ_y}} $$ where $ζ_x$ and $ζ_x$ represent the degrees of the vertices $x$ and $y$, respectively. In this work, we explore the behavior of the $A B C$ index for tree graphs. We establish both lower and upper bounds for the $A B C$ index, expressed in terms of the graph's order and its Roman domination number. Additionally, we characterize the tree structures that correspond to these extremal values, offering a deeper understanding of how the Roman domination number ($RDN$) influences the $A B C$ index in tree graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11850 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremal Values of the Atom-Bond Connectivity Index for Trees with Given Roman Domination Numbers Ali, Waqar Husin, Mohamad Nazri Bin Nadeem, Muhammad Faisal General Mathematics Consider that $\mathbb{G}=(\mathbb{X}, \mathbb{Y})$ is a simple, connected graph with $\mathbb{X}$ as the vertex set and $\mathbb{Y}$ as the edge set. The atom-bond connectivity ($ABC$) index is a novel topological index that Estrada introduced in Estrada et al. (1998). It is defined as $$ A B C(\mathbb{G})=\sum_{xy \in Y(\mathbb{G})} \sqrt{\frac{ζ_x+ζ_y-2}{ζ_x ζ_y}} $$ where $ζ_x$ and $ζ_x$ represent the degrees of the vertices $x$ and $y$, respectively. In this work, we explore the behavior of the $A B C$ index for tree graphs. We establish both lower and upper bounds for the $A B C$ index, expressed in terms of the graph's order and its Roman domination number. Additionally, we characterize the tree structures that correspond to these extremal values, offering a deeper understanding of how the Roman domination number ($RDN$) influences the $A B C$ index in tree graphs. |
| title | Extremal Values of the Atom-Bond Connectivity Index for Trees with Given Roman Domination Numbers |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2411.11850 |