Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

Abstract

For an undirected connected graph G = G(V, E) with vertex set V(G) and edge set E(G), a subset R of V is said to be a resolving in G, if each pair of vertices (say a and b; a ≠ b) in G satisfy the relation d(a, k) ≠ d(b, k), for at least one member k in R. The minimum set R with this resolving property is said to be a metric basis for G, and the cardinality of such set R, is referred to as the metric dimension of G, denoted by dim v (G). In this manuscript, we consider a complex molecular graph of one-heptagonal carbon nanocone (represented by HCN s ) and investigate its metric basis as well as metric dimension. We prove that just three specifically chosen vertices are enough to resolve the molecular graph of HCN s . Moreover, several theoretical as well as applicative properties including comparison have also been incorporated.