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We relate the graph isomorphism problem to the solvability of certain systems of linear equations with nonnegative variables. This version replaces the two previous versions of this paper.

We investigate classical and quantum physics-based algorithms for solving the graph isomorphism problem. Our work integrates and extends previous work by Gudkov et al. (cond-mat/0209112) and by Rudolph (quant-ph/0206068). Gudkov et al. propose an algorithm intended to solve the graph isomorphism problem in polynomial time by mimicking a classical dynamical many-particle process. We show that this algorithm fails to distinguish pairs of non-isomorphic strongly regular graphs, thus providing an infinite class of counterexamples. We also show that the simplest quantum generalization of the algorithm also fails. However, by combining Gudkov et al.'s algorithm with a construction proposed by Rudoph in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested (with up to 29 vertices).

The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph's enumeration with the help of the effective algorithms without their preliminary plotting, etc. The examining of the transformation of the vertex graphs into the edge graph and the opposite operation illustrates the reasons of the appearance of the NP-completeness from the point of view of the graph theory. We suggest that it also illustrates the synchronous possibility and impossibility of the struggle with the NP-completeness.

Quantum annealing is a proposed combinatorial optimization technique meant to exploit quantum mechanical effects such as tunneling and entanglement. Real-world quantum annealing-based solvers require a combination of annealing and classical pre- and post-processing; at this early stage, little is known about how to partition and optimize the processing. This article presents an experimental case study of quantum annealing and some of the factors involved in real-world solvers, using a 504-qubit D-Wave Two machine and the graph isomorphism problem. To illustrate the role of classical pre-processing, a compact Hamiltonian is presented that enables a reduced Ising model for each problem instance. On random N-vertex graphs, the median number of variables is reduced from N^2 to fewer than N lg N and solvable graph sizes increase from N = 5 to N = 13. Additionally, a type of classical post-processing error correction is evaluated. While the solution times are not competitive with classical approaches to graph isomorphism, the enhanced solver ultimately classified correctly every problem that was mapped to the processor and demonstrated clear advantages over the baseline approach. The results shed some light on the nature of real-world quantum annealing and the associated hybrid classical-quantum solvers.

We consider heuristic algorithm for solving graph isomorphism problem. The algorithm based on a successive splitting of the eigenvalues of the matrices which are modifications (to positive defined) of graphs' adjacency matrices. Modification of the algorithm allows to find a solution for Frobenius problem. Formulation of the Frobenius problem is following one. Given a pair of two matrices with the same number of rows and columns. We must find out whether one of the matrix can be acquired from another by permutation of it's rows and strings or not. For example, solution of Frobenius problem can give to us efficient way for decrypting of double permutation cyphers problem for high dimension matrices.

Suppose we are given two graphs on $n$ vertices. We define an observable in the Hilbert space $\Co[(S_n \wr S_2)^m]$ which returns the answer ``yes'' with certainty if the graphs are isomorphic and ``no'' with probability at least $1-n!/2^m$ if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.

This paper presents the novel `uniqueness tree' algorithm, as one possible method for determining whether two finite, undirected graphs are isomorphic. We prove that the algorithm has polynomial time complexity in the worst case, and that it will always detect the presence of an isomorphism whenever one exists. We also propose that the algorithm will equivalently discern the lack of an isomorphism whenever one does not exist, and some initial justifications are given for this proposition, although it cannot yet be rigorously proven. Finally, we present experimental evidence for both the effectiveness and efficiency of the uniqueness tree method, using data gathered from a practical implementation of the algorithm. Some consequences and directions for further research are discussed.

Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.

Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.

We show that the quantum dynamics of interacting and noninteracting quantum particles are fundamentally different in the context of solving a particular computational problem. Specifically, we consider the graph isomorphism problem, in which one wishes to determine whether two graphs are isomorphic (related to each other by a relabeling of the graph vertices), and focus on a class of graphs with particularly high symmetry called strongly regular graphs (SRG's). We study the Green's functions that characterize the dynamical evolution single-particle and two-particle quantum walks on pairs of non-isomorphic SRG's and show that interacting particles can distinguish non-isomorphic graphs that noninteracting particles cannot. We obtain the following specific results: (1) We prove that quantum walks of two noninteracting particles, Fermions or Bosons, cannot distinguish certain pairs of non-isomorphic SRG's. (2) We demonstrate numerically that two interacting Bosons are more powerful than single particles and two noninteracting particles, in that quantum walks of interacting bosons distinguish all non-isomorphic pairs of SRGs that we examined. By utilizing high-throughput computing to perform over 500 million direct comparisons between evolution operators, we checked all tabulated pairs of non-isomorphic SRGs, including graphs with up to 64 vertices. (3) By performing a short-time expansion of the evolution operator, we derive distinguishing operators that provide analytic insight into the power of the interacting two-particle quantum walk.

In this paper, we propose algorithms for the graph isomorphism (GI) problem that are based on the eigendecompositions of the adjacency matrices. The eigenvalues of isomorphic graphs are identical. However, two graphs $ G_A $ and $ G_B $ can be isospectral but non-isomorphic. We first construct a graph isomorphism testing algorithm for friendly graphs and then extend it to unambiguous graphs. We show that isomorphisms can be detected by solving a linear assignment problem. If the graphs possess repeated eigenvalues, which typically correspond to graph symmetries, finding isomorphisms is much harder. By repeatedly perturbing the adjacency matrices and by using properties of eigenpolytopes, it is possible to break symmetries of the graphs and iteratively assign vertices of $ G_A $ to vertices of $ G_B $, provided that an admissible assignment exists. This heuristic approach can be used to construct a permutation which transforms $ G_A $ into $ G_B $ if the graphs are isomorphic. The methods will be illustrated with several guiding examples.

We propose an algorithm for solving of the graph isomorphism problem. Also, we introduce the new class of graphs for which the graph isomorphism problem can be solved polynomially using the algorithm.

We propose an algorithm for solving of the graph isomorphism problem. Also, we introduce the new class of graphs for which the graph isomorphism problem can be solved polynomially using the algorithm.

This paper develops two heuristic algorithms to solve graph isomorphism, using free energy encoding. The first algorithm uses four types of encoding refinement techniques such that every graph can be distinguished by a canonical number computed by the algorithm. The second algorithm injects energy into the graph to conduct individualization such that the correspondence relation between a pair of isomorphic graphs can be found. The core principle behind the two algorithms is encoding discrete structures as real numbers. A large set of experiments demonstrated the effectiveness of our algorithms.

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie in the same G-orbit. Then we attack the Orbit Problem by constructing certain approximate categories C_d(V), d=1,2,3,... whose objects include the elements of V. We show that v_1 and v_2 are not in the same orbit by showing that they are not isomorphic in the category C_d(V) for some d. For every d this gives us an algorithm for isomorphism testing. We will show that the WL-algorithms reduce to our algorithms, but that our algorithms cannot be reduced to the WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can distinguish the Cai-Furer-Immerman graphs in polynomial time.

We propose a novel method using a quantum annealer -- an analog quantum computer based on the principles of quantum adiabatic evolution -- to solve the Graph Isomorphism problem, in which one has to determine whether two graphs are isomorphic (i.e., can be transformed into each other simply by a relabeling of the vertices). We demonstrate the capabilities of the method by analyzing several types of graph families, focusing on graphs with particularly high symmetry called strongly regular graphs (SRG's). We also show that our method is applicable, within certain limitations, to currently available quantum hardware such as "D-Wave One".

An isomorphism between two graphs is a bijection between their vertices that preserves the edges. We consider the problem of determining whether two finite undirected weighted graphs are isomorphic, and finding an isomorphism relating them if the answer is positive. In this paper we introduce effective probabilistic linear programming (LP) heuristics to solve the graph isomorphism problem. We motivate our heuristics by showing guarantees under some conditions, and present numerical experiments that show effectiveness of these heuristics in the general case.

The graph isomorphism problem is considered. We assign modified characteristic polynomials for graphs and reduce the graph isomorphism problem to the following one. It is required to find out, is there such an enumeration of the graphs vertices that the polynomials of the graphs are equal. We present algorithms for the graph isomorphism problem that use the reduction. We prove the propositions that justify the possibility of a numerical realization of the algorithms for the general case of the graph isomorphism problem. The algorithms perform equality checking of graphs modified $n$-variables characteristic polynomials. We show that probability of obtaining a wrong solution of the graph isomorphism problem using recursive modification of the algorithm is negligible if the algorithm parameter is sufficiently large. In the course of its implementation, the algorithm checks the equality of the graphs modified characteristic polynomials in predefined points. For $n$-vertices graph, the polynomial has $2^n$ coefficients so its value in some point cannot be evaluated directly for large enough $n$. We show that we may check the equality of the polynomials in predefined points without direct evaluation of the polynomials values in these points. We prove that, for the graphs on $n$ vertices, it is required $O(n^4)$ elementary machine operations and it is requred machine numbers with mantissa's length $O(n^2)$ to check equality of the graphs polynomials values in predefined points. In general, it needs an exponential time to solve the $GI$ instance using the presented approach, but in practice, it is efficient even for compuationally hard instances of the graph isomorphism problem.

This paper provides a new framework to study graph structure. It enables us to define graph orthogonality and a new metrics on graphs. Furthermore, it enables us to define the coordinate representation of graphs with respect to an ordered set of graphs, which benefits us in the graph isomorphism problem. If any graph of set $A$ has a unique coordinates with respect to a graph set $ B$, then, we call $B$ is a basis for the graphs set $A$. Having a basis, any graph finds a unique coordinates. Thus, the graph isomorphism problem equals to comparison of the coordinates. This fact provides a formal approach to study the computational complexity of the graph isomorphism problem, i.e. finding a suitable basis. We have shown that graphs on at most 3log(n) vertices make a basis for almost every $n$-vertex graphs. This fact, easily, results that graph isomorphism problem can be solved in $\exp(6\log^2(n))$ time for almost every pair of $n$-vertex graphs.

This paper provides a new framework to study graph structure. It enables us to define graph orthogonality and a new metrics on graphs. Furthermore, it enables us to define the coordinate representation of graphs with respect to an ordered set of graphs, which benefits us in the graph isomorphism problem. If any graph of set $A$ has a unique coordinates with respect to a graph set $ B$, then, we call $B$ is a basis for the graphs set $A$. Having a basis, any graph finds a unique coordinates. Thus, the graph isomorphism problem equals to comparison of the coordinates. This fact provides a formal approach to study the computational complexity of the graph isomorphism problem, i.e. finding a suitable basis. We have shown that graphs on at most 3log(n) vertices make a basis for almost every $n$-vertex graphs. This fact, easily, results that graph isomorphism problem can be solved in $\exp(6\log^2(n))$ time for almost every pair of $n$-vertex graphs.

Given two graphs $G_1$ and $G_2$ on $n$ vertices each, we define a graph $G$ on vertex set $V_1\times V_2$ and the edge set as the union of edges of $G_1\times \bar{G_2}$, $\bar{G_1}\times G_2$, $\{(v,u'),(v,u"))(|u',u"\in V_2\}$ for each $v\in V_1$, and $\{((u',v),(u",v))|u',u"\in V_1\}$ for each $v\in V_2$. We consider the completely-positive Lov\'asz $\vartheta$ function, i.e., $cp\vartheta$ function for $G$. We show that the function evaluates to $n$ whenever $G_1$ and $G_2$ are isomorphic and to less than $n-1/(4n^4)$ when non-isomorphic. Hence this function provides a test for graph isomorphism. We also provide some geometric insight into the feasible region of the completely positive program.

In this paper we introduce a novel polynomial-time algorithm to compute graph invariants based on the modified random walk idea on graphs. However not proved to be a full graph invariant by now, our method gives the right answer for the graph instances other well-known methods could not compute (such as special Furer Gadgets and point-line incidence graphs of finite projective planes of higher degrees