An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops.

Naveed Ahmed Azam,Hiroshi Nagamochi,Aleksandar Shurbevski

doi:10.3390/e22090923

Abstract

Graph enumeration with given constraints is an interesting problem considered to be one of the fundamental problems in graph theory, with many applications in natural sciences and engineering such as bio-informatics and computational chemistry. For any two integers and , we propose a method to count all non-isomorphic trees with n vertices, self-loops, and no multi-edges based on dynamic programming. To achieve this goal, we count the number of non-isomorphic rooted trees with n vertices, self-loops and no multi-edges, in time and space, since every tree can be uniquely viewed as a rooted tree by either regarding its unicentroid as the root, or in the case of bicentroid, by introducing a virtual vertex on the bicentroid and assuming the virtual vertex to be the root. By this result, we get a lower bound and an upper bound on the number of tree-like polymer topologies of chemical compounds with any “cycle rank”.

Highlights

Counting and generation of discrete objects are two fundamental problems in combinatorial mathematics and have many applications in the fields of natural science and engineering, such as computational chemistry and bioinformatics
The computation is performed by following a computation tree, and the required solutions are attained at the leaves of the computation tree
We develop a method to compute for any two integers n ≥ 1 and ∆ ≥ 0, the size h(n, ∆) of a maximal set H(n, ∆) of mutually non-isomorphic rooted trees with n vertices and ∆ self-loops; i.e., we are interested in the following problem: Counting Problem

Summary

Introduction

Counting and generation of discrete objects are two fundamental problems in combinatorial mathematics and have many applications in the fields of natural science and engineering, such as computational chemistry and bioinformatics. The well-known Polya’s enumeration theorem [1,2] is used for counting all distinct objects The idea of this method is to use the cyclic index of the group of symmetries of the underlying object to develop a generating function, which is used to count all possible objects. The problem of counting and generation of chemical compounds can be viewed as the problem of enumerating graphs with given constraints. Enumeration of restricted chemical compounds with specialized tools is more efficient than with the tools which use general graph structures. Unlike branching algorithms and Polya’s theorem, the main advantage of using the DP is that we can count all non-isomorphic structures without their generation and calculation of their group of symmetries.

Preliminaries

Counting Tree-Like Graphs with a Given Number of Vertices and Self-Loops

Conclusions