2024/10/31 by Waqar Ali, Mohamad Nazri Husin, Ali, Waqar +4
Computer Science · Mathematics · #Advanced Graph Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Graph theory and applications #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2411.11851
openalex publication_date 2024/10/31 · openalex created_date 2024/11/21 · openalex updated_date 2026/07/28
Let \mathbbG = (V, E) be a simple connected graph, where V and E denote the vertex and edge sets, respectively. The first Zagreb index is defined as M1(\mathbbG) = ∑v ∈ V ζ_\mathbbG(v)2, while the second Zagreb index is given by M2(\mathbbG) = ∑uv ∈ E ζ_\mathbbG(u) ζ_\mathbbG(v), where ζ_\mathbbG(v) represents the degree of vertex v. Another notable degree-based invariant is the atom-bond connectivity (ABC) index, introduced in chemical graph theory, and defined by ABC(\mathbbG) = ∑uv ∈ E √\fracζ_\mathbbG(u) + ζ_\mathbbG(v) - 2ζ_\mathbbG(u) ζ_\mathbbG(v). A fundamental graph parameter, the metric dimension, refers to the minimum number of vertices in a resolving set that uniquely distinguishes all other vertices based on distances. In this work, we investigate the influence of metric dimension on the Zagreb and ABC indices within the class of trees. We derive sharp bounds-both upper and lower for M1 and M2, and provide an upper bound for the ABC index, all expressed in terms of the tree's order and its metric dimension. Furthermore, we identify the extremal tree structures that attain these bounds. These findings underscore the role of metric dimension in shaping topological descriptors and contribute both to theoretical graph analysis and practical applications in molecular chemistry.