This paper deals with graph invariants and stabilization procedures. We consider colored graphs and their automorphisms and we discuss the isomorphism problem for such graphs. Various global and local isomorphism invariants are introduced. We study canonical numberings, invariant partitions, stable and equitable partitions and algorithms for stabilizing partitions.

In this paper, we show the existence of a polynomial time graph isomorphism algorithm for a restricted class of graphs which are not triangle-free graphs. We consider graphs that have atleast 2 vertices in all rearranged sub-graphs. An Individualization-Refinement procedure(1 Dimensional Weisfeiler-Lehman Method) is used to rearrange one graph which splits the graph into disjoint vertex sets. T...

Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in TC1 ⊆ NC2. If no coun...

The subgraph isomorphism problem has various important applications, although it is generally NP-complete and difficult to solve. This paper examines the feasibility of a data dependent circuit for the subgraph isomorphism problem, which is particularly suitable for FPGA implementation. For graphs of 32 vertices, the average logic scale of data dependent circuits is only 5% of the corresponding...

For similarity measures of labeled and unlabeled graphs, we study the complexity of the graph isomorphism problem for pairs of input graphs which are close with respect to the measure. More precisely, we show that for every fixed integer k we can decide in quadratic time whether a labeled graph G can be obtained from another labeled graphH by relabeling at most k vertices. We extend the algorit...

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are respon...