### Journals Information

**
Mathematics and Statistics Vol. 11(6), pp. 953 - 959 DOI: 10.13189/ms.2023.110610 Reprint (PDF) (342Kb) **

## On The Metric Dimension for The Line Graphs of Hammer and Triangular Benzene Structures

**R. Nithya Raj ^{1}, R. Sundara Rajan ^{1}, Haewon Byeon ^{2}^{,*}, CT. Nagaraj ^{3}, G. Kokila ^{4}**

^{1}Department of Mathematics, Hindustan Institute of Technology and Science, Chennai, India

^{2}Department of AI Big Data, Inje University, Gimhae, 50834, Republic of Korea

^{3}Department of Mathematics, Sree Sevugan Annamalai College, Devakottai-630303, Tamilnadu, India

^{4}Department of Mathematics, Unique College of Arts and Science, Karapattu, Uthangarai - 63520, India

**ABSTRACT**

The metric dimension of a chemical graph is a fundamental parameter in the study of molecular structures and their properties. This metric dimension is a numerical measure of the smallest set of atoms required to uniquely determine the location of all other atoms within the molecule. In this abstract, we explore the concept of metric dimension in chemical graphs, discussing its theoretical foundations and its applications in various fields such as navigation, network theory, drug design, optimization, pattern recognition, and other related fields computational chemistry, and material science. Understanding the metric dimension of chemical graphs enables the identification of crucial atoms or bonds that significantly impact the properties and behavior of molecules, aiding in the design of more effective drugs, catalysts, and materials. Finding the metric dimension of any given graph poses a computational challenge classified within the NP-complete problem category. A group of nodes, denoted as , is regarded as a locating set if, every pair of nodes and within the graph , there is a minimum of one node in such a way that the separation between and is not the same as the separation between and . The metric dimension is represented by and corresponds to the minimum size of a locating set for . The primary objective of this effort is to establish the proof that, for , the metric dimensions of the line network for the Hammer and triangular benzene structures are 2 and 3, respectively. We also established the existence of a constant metric dimension for this class of line graphs, which includes Hammer and triangular benzene structures.

**KEYWORDS**

Metric Basis, Metric Dimension, Hammer Graph, Triangular Benzenoid Structure

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] R. Nithya Raj , R. Sundara Rajan , Haewon Byeon , CT. Nagaraj , G. Kokila , "On The Metric Dimension for The Line Graphs of Hammer and Triangular Benzene Structures," Mathematics and Statistics, Vol. 11, No. 6, pp. 953 - 959, 2023. DOI: 10.13189/ms.2023.110610.

(b). APA Format:

R. Nithya Raj , R. Sundara Rajan , Haewon Byeon , CT. Nagaraj , G. Kokila (2023). On The Metric Dimension for The Line Graphs of Hammer and Triangular Benzene Structures. Mathematics and Statistics, 11(6), 953 - 959. DOI: 10.13189/ms.2023.110610.