Abstract
In chemistry, graphs are commonly used to show the structure of chemical compounds, with nodes and edges representing the atom and bond types, respectively. Edge resolving set λ e is an ordered subset of nodes of a graph C , in which each edge of C is distinctively determined by its distance vector to the nodes in λ . The cardinality of a minimum edge resolving set is called the edge metric dimension of C . An edge resolving set L e , f of C is fault-tolerant if λ e , f ∖ b is also an edge resolving set, for every b in λ e , f . Resolving set allows obtaining a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. In this paper, we determine the exact edge metric and fault-tolerant edge metric dimension of benzenoid tripod structure and proved that both parameters are constant.
Highlights
Mathematical chemistry has recently presented a wide range of approaches to understanding the chemical structures that underpin existing chemical theories and developing and exploring new mathematical models of chemical phenomena and applying mathematical concepts and processes to chemistry
After physicists revealed in the first few years of the twentieth century that the key features of chemical compounds can be predicted using quantum theory approaches, a significant need for mathematics in chemistry arose. e main driving force that drove the mathematics and its concepts into chemistry laboratories was the realization that chemistry cannot be comprehended without knowledge of quantum physics, including its complicated mathematical instruments
Chemical graph theory is a branch of mathematical science that is used to characterise the structural properties of molecules, processes, crystals, polymers, clusters, and other objects. e vertex of a chemical graph theory might be an electron, an atom, a molecule, a collection of atoms, intermediates, orbitals, and many other things
Summary
Mathematical chemistry has recently presented a wide range of approaches to understanding the chemical structures that underpin existing chemical theories and developing and exploring new mathematical models of chemical phenomena and applying mathematical concepts and processes to chemistry. Assume that C is an associated graph of chemical structure/network, whose vertex/node set are denoted by symbol N(C) or N, while B(C) or B is the edge/bond set, and the shortest distance between two bonds b1, b2 ∈ N(C) is denoted by Sb1,b2 and calculated by counting the number of bonds while moving through the b1 − b2 path. Assuming a subset of selected nodes λe, if the position p(e|λe) of each edge e is unique of a graph, λe is called as edge metric resolving set and dime(C) is the minimum count of members of λe, called as edge metric dimension. If dime(C) is the edge metric dimension, dime(C) 1, iff C is a path Pn
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