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

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.