We study the time complexity of McKay's algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered automorphisms and by hashing partial information of vertex labelings. In practice, the algorithm is implemented in the nauty package. We obtain colorings of Furer's graphs that allow the algorithm to compute their canonical forms in polynomial time. We then prove an exponential lower bound of the algorithm for connected 3-regular graphs of color-class size 4 using Furer's construction. We conducted experiments with nauty for these graphs. Our experimental results also indicate the same exponential lower bound.

Despite the significant number of isomorphism algorithms presented in the literature, till now no efforts have been done for characterizing their performance. Consequently, it is not clear how the behavior of those algorithms varies as the type and the size of the graphs to be matched varies in case of real applications. In this paper we present a benchmarking activity for characterizing the performance of a bunch of algorithms for exact graph isomorphism. To this purpose we use a large database containing 10,000 couples of isomorphic graphs with different topologies (regular graphs, randomly connected graphs, bounded valence graph), enriched with suitably modified versions of them for simulating distortions occurring in real cases. The size of the considered graphs ranges from a few nodes to about 1000 nodes. 1.

0. How to use this Guide.

The MOLGEN Chemical Identifier MOLGEN-CID is a software module freely accessible via the Internet. For a molecule or graph entered in molfile format it produces, by a canonical renumbering procedure, a canonical molfile and a unique character string that is easily compared by computer to a similar string. The mode of operation of MOLGEN-CID is detailed and visualized with examples.

Graphs are an extremely general and powerful data structure. In pattern recognition and computer vision, graphs are used to represent patterns to be recognized or classified. Detection of maximum common subgraph (MCS) is useful for matching, comparing and evaluate the similarity of patterns. MCS is a well known NP-complete problem for which optimal and suboptimal algorithms are known from the literature. Nevertheless, until now no effort has been done for characterizing their performance. The lack of a large database of graphs makes the task of comparing the performance of different graph matching algorithms difficult, and often the selection of an algorithm is made on the basis of a few experimental results available. In this paper, three optimal and well-known algorithms for maximum common subgraph detection are described. Moreover a large database containing various categories of pairs of graphs (e.g. random graphs, meshes, bounded valence graphs), is presented, and the performance

The PageRank algorithm perturbs the adjacency matrix defined by a set of web pages and hyperlinks such that the resulting matrix is positive and row-stochastic. Applying the Perron-Frobenius theorem establishes that the eigenvector associated with the dominant eigenvalue exists and is unique. For some graphs, the PageRank algorithm may yield a canonical isomorph. We propose a ranking method based on the matrix inverse. Since the inverse may not exist, we apply two isomorphismpreserving perturbations, based on the signless Laplacian, to ensure that the resulting matrix is diagonally dominant. By applying the Gershgorin Circle theorem, we know this matrix must have an inverse, namely, a set of vectors unique up to isomorphism. We concatenate sorted rows of the inverse with its unsorted rows, lexicographically sort on the concatenated matrix, and apply the ranking as an induced permutation on the input adjacency matrix. This preliminary report shows IsoRank identifies most random graphs and always terminates in polynomial time, illustrated by the execution run times for a small set of graphs. IsoRank has been applied to dense graphs of as many as 4,000 vertices.

Abstract not found