Edge dependent fault-tolerance in certain carbon-based crystal structures

Abstract

The field of chemical graph theory encompasses interdisciplinary studies that utilize mathematical methods in exploring the attributes and arrangement of chemical compounds. Chemical graphs, in which atoms are vertices and bonds are edges, offer influential mathematical representations for portraying the molecular structures of chemical compounds. A smallest subset of vertices of a graph from which the vector of distances to each vertex of the graph is unique is known as metric basis for that graph and cardinality of a metric basis is known as metric dimension of that graph. In this paper, an important variant of metric dimension known as fault-tolerant edge metric dimension (FTEMD) is taken into consideration and computed it for most important possible allotrope of carbon family, known as crystal cubic carbon (denoted by G(n)). We prove that the FTEMD value of the molecular graph of crystal cubic carbon is unbounded for large n. Also, the comparison between several variants of metric dimension and FTEMD for the molecular graph G(n) has also been incorporated in the paper.