Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank

Naveed Ahmed Azam, Aleksandar Shurbevski, Hiroshi Nagamochi
2020 Entropy  
Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating
more » ... phs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integer n≥1, it generates all required graphs with n vertices in O(n) time per graph and O(n) space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.
doi:10.3390/e22111295 pmid:33287063 fatcat:jcfnjh3rlzclrfd5t2x55nb5cu