We characterize the distributions of short cycles in a large metabolic network previously shown to have small world characteristics and a power law degree distribution. Compared with three classes of random networks, including Erdős–Rényi random graphs and synthetic small world networks of the same connectivity, both the metabolic network and models for the chemical reaction networks of planetary atmospheres have a particularly large number of triangles and a deficit in large cycles. Short cycles reduce the length of detours when a connection is clipped, so we propose that long cycles in metabolism may have been selected against in order to shorten transition times and reduce the likelihood of oscillations in response to external perturbations.
Abstract. Ring information is a large part of the structural topology used to identify and characterize molecular structures. It is hence of crucial importance to obtain this information for a variety of tasks in computational chemistry. Many different approaches for "ring perception", i.e., the extraction of cycles from a molecular graph, have been described. The chemistry literature on this topic, however, reports a surprisingly large number of incorrect statements about the properties of chemically relevant ring sets and, in particular, about the mutual relationships of different sets of cycles in a graph. In part these problems seem to have arisen from a sometimes rather idiosyncratic terminology for notions that are fairly standard in graph theory. In this contribution we translate the definitions of concepts such as the Smallest Set of Smallest Rings, Essential Set of Essential Rings, Extended Set of Smallest Rings, Set of Smallest Cycles at Edges, Set of Elementary Rings, K-rings, and β-rings into a more widely-used mathematical language. We then outline the basic properties of different cycle sets and provide numerous counterexamples to incorrect claims in the published literature. These counterexamples may have a serious practical impact because at least some of them are molecular graphs of well-known molecules. As a consequence, we propose a catalogue of desirable properties for chemically useful sets of rings.
The cycle space of a strongly connected graph has a basis consisting of directed circuits. The concept of relevant circuits is introduced as a generalization of the relevant cycles in undirected graphs. A polynomial time algorithm for the computation of a minimum weight directed circuit basis is outlined.
The set ${\cal R}$ of relevant cycles of a graph $G$ is the union of its minimum cycle bases. We introduce a partition of ${\cal R}$ such that each cycle in a class ${\cal W}$ can be expressed as a sum of other cycles in ${\cal W}$ and shorter cycles. It is shown that each minimum cycle basis contains the same number of representatives of a given class ${\cal W}$. This result is used to derive upper and lower bounds on the number of distinct minimum cycle bases. Finally, we give a polynomial-time algorithm to compute this partition.
Given an undirected graph G(V, E) and a vertex subset U ⊆ V the Uspace is the vector space over GF(2) spanned by the paths with end-points in U and the cycles in G(V, E). We extend Vismara's algorithm to the computation of the union of all minimum length bases of the U-space. While the size distribution of subgraphs is the same in all minimum length bases, the number of cycles and paths may differ.
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