Bounds on Atom-Bond Connectivity and Zagreb Indices in Trees with a Given Metric Dimension

Abstract

Let G=(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 H1(G)=∑ν∈VdegG(ν)2. Meanwhile, the second Zagreb index is given by H2(G)=∑ν1ν2∈EdegG(ν1)degG(ν2), where degG(ν) represents the degree of vertex ν. Another notable degree-based invariant is the Atom-Bond Connectivity (ABC) index, introduced in chemical graph theory, and defined by: ABC(G)=∑ν1ν2∈EdegG(ν1)+degG(ν2)−2degG(ν1)degG(ν2)−−−−−−−−−−−−−−−−−−−−√. A fundamental graph parameter, the Metric Dimension (MetDim), 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 MetDim on the Zagreb and ABC indices within the class of trees. Accordingly, we derive sharp bounds-both upper and lower for H1(T) and H2(T), and provide an upper bound for the ABC(T) index. All are expressed in terms of the tree's order and its MetDim. Additionally, we recognize the extremal tree structures that attain these extreme values. Overall, these findings underscore the role of MetDim in shaping topological descriptors and contribute to both theoretical graph analysis and practical applications in molecular chemistry.