Graph isomorphism and multivariate graph spectrum

We derive structural constraints on the automorphism groups of strongly regular (s.r.) graphs, giving a surprisingly strong answer to a decades-old problem, with tantalizing implications to testing isomorphism of s.r. graphs, and raising new combinatorial challenges. S.r. graphs, while not believed to be Graph Isomorphism (GI) complete, have long been recognized as hard cases for GI, and, in this author's view, present some of the core difficulties of the general GI problem. Progress on the complexity of testing their isomorphism has been intermittent (Babai 1980, Spielman 1996, BW & CST (STOC'13) and BCSTW (FOCS'13)), and the current best bound is exp(O(n1/5)) (n is the number of vertices). Our main result is that if X is a s.r. graph then, with straightforward exceptions, the degree of the largest alternating group involved in the automorphism group Aut(X) (as a quotient of a subgroup) is O((ln n)2ln ln n). (The exceptions admit trivial linear-time GI testing.) The design of isomorphism tests for various classes of structures is intimately connected with the study of the automorphism groups of those structures. We include a brief survey of these connections, starting with an 1869 paper by Jordan on trees. In particular, our result amplifies the potential of Luks's divide-and-conquer methods (1980) to be applicable to testing isomorphism of s.r. graphs in quasipolynomial time. The challenge remains to find a hierarchy of combinatorial substructures through which this potential can be realized. We expect that the generality of our result will help in this regard; the result applies not only to s.r. graphs but to all graphs with strong spectral expansion and with a relatively small number of common neighbors for every pair of vertices. We state a purely mathematical conjecture that could bring us closer to finding the right kind of hierarchy. We also outline the broader GI context, and state conjectures in terms of primitive coherent configurations. These are generalizations of s.r. graphs, relevant to the general GI problem. Another consequence of the main result is the strongest argument to date against GI-completeness of s.r. graphs: we prove that no polynomial-time reduction of GI to isomorphism of s.r. graphs is possible. All known reductions between isomorphism problems of various classes of structures fit into our notion of categorical reduction. The proof of the main result is elementary; it is based on known results in spectral graph theory and on a 1987 lemma on permutations by Akos Seress and the author.

Detection of graph isomorphism (GI) has been widely used in many fields in science and engineering. Currently, a potential application of GI detection could be in molecular structure design for microelectromechanical systems and nano-systems. In this paper, we discuss the relationship between graphs and their eigenvalues as well as unique eigenvectors. We prove that the graphs having all distinct eigenvalues are isomorphic if and only if they have the same graph spectrum and the equivalent eigenvectors. The graphs having coincident eigenvalues might be isomorphic if they have the same graph spectrum and the equivalent unique eigenvectors. Further, a convergent recursive procedure is given to subdivide a group-to-group mapping once appeared in the graphs having coincident eigenvalues to seek potential one-to-one mappings so as to determining if the graphs are isomorphic.

The graph isomorphism problem is to check if two given graphs are isomorphic. Graph isomorphism is a well studied problem and numerous algorithms are available for its solution. In this paper we present algorithms for graph isomorphism that employ the spectra of graphs. An open problem that has fascinated many a scientist is if there exists a polynomial time algorithm for graph isomorphism. Though we do not solve this problem in this paper, the algorithms we present take polynomial time. These algorithms have been tested on a good collection of instances. However, we have not been able to prove that our algorithms will work on all possible instances. In this paper, we also give a new construction for cospectral graphs.

INTRODUCTION. ELEMENTS OF GRAPH THEORY. The Definition of a Graph. Isomorphic Graphs and Graph Automorphism. Walks, Trails, Paths, Distances and Valencies in Graphs. Subgraphs. Regular Graphs. Trees. Planar Graphs. The Story of the Koenigsberg Bridge Problem and Eulerian Graphs. Hamiltonian Graphs. Line Graphs. Vertex Coloring of a Graph. CHEMICAL GRAPHS. The Concept of a Chemical Graph. Molecular Topology. Huckel Graphs. Polyhexes and Benzenoid Graphs. Weighted Graphs. GRAPH-THEORETICAL MATRICES. The Adjacency Matrix. The Distance Matrix. THE CHARACTERISTIC POLYNOMIAL OF A GRAPH. The Definition of the Characteristic Polynomial. The Method of Sachs for Computing the Characteristic Polynomial. The Characteristic Polynomials of Some Classes of Simple Graphs. The Le Verrier-Faddeev-Frame Method for Computing the Characteristic Polynomial. TOPOLOGICAL ASPECTS OF HUECKEL THEORY. Elements of Huckel Theory. Isomorphism of Huckel Theory and Graph Spectral Theory. The Huckel Spectrum. Charge Densities and Bond Orders in Conjugated Systems. The Two-Color Problem in Huckel Theory. Eigenvalues of Linear Polyenes. Eigenvalues of Annulenes. Eigenvalues of Moebius Annulenes. A Classification Scheme for Monocyclic Systems. Total p-Electron Energy. TOPOLOGICAL RESONANCE ENERGY. Huckel Resonance Energy. Dewar Resonance Energy. The Concept of Topological Resonance Energy. Computation of the Acyclic Polynomial. Applications of the TRE Model. ENUMERATION OF KEKULE VALENCE STRUCTURES. The Role of Kekule Valence Structures in Chemistry. The Identification of Kekule Systems. Methods for the Enumeration of Kekule Structures. The Concept of Parity of Kekule Structures. THE CONJUGATED-CIRCUIT MODEL. The Concept of Conjugated Circuits. The p-Resonance Energy Expression. Selection of the Parameters. Computational Procedure. Applications of the Conjugated-Circuit Model. Parity of Conjugated Circuits. TOPOLOGICAL INDICES. Definitions of Topological Indices. The Three-Dimensional Wiener Number. ISOMER ENUMERATION. The Cayley Generation Functions. The Henze-Blair Approach. The Polya Enumeration Method. The Enumeration Method Based on the N-Tuple Code.

We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.

The spectral density of a graph is a key concept for quantitatively characterizing empirical networks. It has many applications, including community detection, graph signal processing, spectral embedding, network evolution, brain network analysis, and random graph modeling. The graph’s spectral density is also crucial in developing statistical methods for graphs, such as model selection and comparative testing. Despite its broad applicability, a complete understanding of the relationship between a graph’s spectral density and structure remains elusive. To advance our understanding of the relationship between graph spectra and their structure, we introduce a vertex-wise decomposition of the graph’s spectral density, allowing us to determine each vertex’s contribution to specific eigenvalues. We show that the decomposition of distinct isospectral graphs (graphs with identical spectra) can be distinguished by the vertex-wise graph spectra, showing that the proposed new quantities are finer invariants between isomorphic graphs. Finally, we apply these insights to analyze chemical molecules and identify genes associated with normal versus tumoral breast gene interaction networks.

For each p\geq 1 , the star automaton group \mathcal{G}_{S_p} is an automaton group which can be defined starting from a star graph on p+1 vertices. We study Schreier graphs associated with the action of the group \mathcal{G}_{S_p} on the regular rooted tree T_{p+1} of degree p+1 and on its boundary \partial T_{p+1} . With the transitive action on the n -th level of T_{p+1} is associated a finite Schreier graph \Gamma^{p}_{n} , whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as the limits of the sequence \{\Gamma_{n}^{p}\}_{n\geq 1} in the Gromov–Hausdorff topology. We obtain an explicit description of the spectrum of the graphs \{\Gamma_{n}^{p}\}_{n\geq 1} . Then, by using amenability of \mathcal{G}_{S_p} , we prove that the spectrum of each infinite Schreier graph is the union of a Cantor set of zero Lebesgue measure, which is the Julia set of the quadratic map f_{p}(z) = z^{2}-2(p-1)z -2p , and a countable collection of isolated points supporting the Kesten–Neumann–Serre spectral measure. We also give a complete classification of the infinite Schreier graphs up to isomorphism of unrooted graphs, showing that they may have 1 , 2 or 2p ends, and that the case of 1 end is generic with respect to the uniform measure on \partial T_{p+1} .

Signal processing on graphs using adjacency matrix (as opposed to more traditional graph Laplacian) results in an algebraic framework for graph signals and shift invariant filters. This can be seen as an example of the algebraic signal processing theory. In this study, the authors examine the concepts of homomorphism and isomorphism between two graphs from a signal processing point of view and refer to them as GSP isomorphism and GSP homomorphism, respectively. Collectively, they refer to these concepts as structure preserving maps (SPMs). The fact that linear combination of signals and linear transforms on signals are meaningful operations has implications on the GSP isomorphism and GSP homomorphism, which diverges from the topological interpretations of the same concepts (i.e. graph isomorphism and graph homomorphism). When SPMs exist between two graphs, signals and filters can be mapped between them while preserving spectral properties. They examine conditions on adjacency matrices for such maps to exist. They also show that isospectral graphs form a special case of GSP isomorphism and that GSP isomorphism and GSP homomorphism is intrinsic to resampling and downsampling process.

Recognizing useful modular robot configurations composed of hundreds of modules is a significant challenge. Matching a new modular robot configuration to a library of known configurations is essential in identifying and applying control schemes. We present three different algorithms to address the problem of (a) matching and (b) mapping new robot configurations onto a library of known configurations. The first method solves the problem using graph isomorphisms and can identify configurations that share the same underlying graph structure, but have different port connections amongst the modules. The second approach compares graph spectra of configuration matrices to find a permutation matrix that maps a given configuration to a known one. The third algorithm exploits the unique structure of the problem for the particular robots used in our research to achieve impressive gains in performance and speed over existing techniques, especially for larger configurations. With these three algorithms, this paper presents novel solutions to the problem of configuration recognition and sheds light on theoretical and practical issues for long-term advances in this important area of modular robotics. Results and examples are provided to compare the performance of the three algorithms and discuss their relative advantages.

Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated through the construction of graphs which are used as tools in computer science. Conversely, graph theory has also helped to characterize certain algebraic properties of abstract algebraic structures. In this survey, we highlight the rich interplay between the two topics viz groups and power graphs from groups. In the last decade, extensive contribution has been made towards the investigation of power graphs. Our main motive is to provide a complete survey on the connectedness of power graphs and proper power graphs, the Laplacian and adjacency spectrum of power graph, isomorphism, and automorphism of power graphs, characterization of power graphs in terms of groups. Apart from the survey of results, this paper also contains some new material such as the contents of Section 2 (which describes the interesting case of the power graph of the Mathieu group M11) and Section 6.1 (where conditions are discussed for the reduced power graph to be not connected). We conclude this paper by presenting a set of open problems and conjectures on power graphs.

Gain total graphs and their spectra via G-phases and group representations

In this research paper, it is proved that two arbitrary graphs are isomorphic only if the quadratic forms associated with the two adjacency matrices are same(upto reordering the monomials). Based on the proof, a polynomial time algorithm is designed for testing necessary condition for isomorphism of graph (i.e. effectively deciding whether two graphs are isomorphic). The algorithm requires O(N 3 ) comparison operations. Also, a polynomial time algorithm for checking necessary condition on whether two graphs are isomorphic is designed under the condition that the associated adjacency matrices are non-singular and are related through a symmetric permutation matrix. The algorithms are essentially based on linear algebraic concepts related to graphs. Also, some new results in spectral graph theory are discussed.

On the spectral Ádám property for circulant graphs

Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph theory can be used to map these graphs onto lower dimensional spaces and match shapes by aligning their embeddings in virtue of their invariance to change of pose. Classical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formulation that finds the best alignment between two congruent K-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with histograms, and the retained set provides a smart initialization for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of embeddings under orthogonal transformations; the registration is solved using the framework of unsupervised clustering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topology, shape variations and different sampling densities.

GBP: Graph convolutional network embedded in bilinear pooling for fine-grained encoding