Non-isomorphic Graphs Research Articles

Two-particle quantum walks: Entanglement and graph isomorphism testing

We study discrete-time quantum walks on the line and on general undirected graphs with two interacting or noninteracting particles. We introduce two simple interaction schemes and show that they both lead to a diverse range of probability distributions that depend on the correlations and relative phases between the initial coin states of the two particles. We investigate the characteristics of these quantum walks and the time evolution of the entanglement between the two particles from both separable and entangled initial states. We also test the capability of two-particle discrete-time quantum walks to distinguish nonisomorphic graphs. For strongly regular graphs, we show that noninteracting discrete-time quantum walks can distinguish some but not all nonisomorphic graphs with the same family parameters. By incorporating an interaction between the two particles, all nonisomorphic strongly regular graphs tested are successfully distinguished.

One-regular graphs of square-free order of prime valency

A graph is one-regular if its automorphism group acts regularly on the set of arcs of the graph. Marušič and Pisanski [D. Marušič and T. Pisanski, Symmetries of hexagonal graphs on the torus, Croat. Chemica Acta 73 (2000) 969–981] classified one-regular Cayley graphs on a dihedral group of valency 3, and Kwak et al. [J.H. Kwak, Y.S. Kwon, J.M. Oh, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, J. Combin. Theory B 98 (2008) 585–598] classified those of valency 5. In this paper one-regular Cayley graphs on a dihedral group of any prime valency are classified and enumerated. It is shown that for an odd prime q , there exists a q -valent one-regular Cayley graph on the dihedral group of order 2 m if and only if m = q t p 1 e 1 p 2 e 2 ⋯ p s e s ≥ 13 , where t ≤ 1 , s ≥ 1 , e i ≥ 1 and p i ’s are distinct primes such that q ∣ ( p i − 1 ) . There are exactly ( q − 1 ) s − 1 non-isomorphic such graphs for a given order. Consequently, one-regular cyclic Haar graphs of prime valency are classified and enumerated. Furthermore, it is shown that every q -valent one-regular graph of square-free order is a Cayley graph on a dihedral group, and as a result, q -valent one-regular graphs of square-free order are classified and enumerated.

Isomorphism Testing via Polynomial-Time Graph Extensions

This paper deals with algorithms for detecting graph isomorphism (GI) properties. The GI literature consists of numerous research directions, from highly theoretical studies (e.g. defining the GI complexity class) to very practical applications (pattern recognition, image processing). We first present the context of our work and provide a brief overview of various algorithms developed in such disparate contexts. Compared to well-known NP-complete problems, GI is only rarely tackled with general-purpose combinatorial optimization techniques; however, classical search algorithms are commonly applied to graph matching (GM). We show that, by specifically focusing on exploiting isomorphism properties, classical GM heuristics can become very useful for GI. We introduce a polynomial graph extension procedure that provides a graph coloring (labeling) capable of rapidly guiding a simple-but-effective heuristic toward the solution. The resulting algorithm (GI-Ext) is quite simple, very fast and practical: it solves GI within a time in the region of O(|V|3) for numerous graph classes, including difficult (dense and regular) graphs with up to 20.000 vertices and 200.000.000 edges. GI-Ext can compete with recent state-of-the-art GI algorithms based on well-established GI techniques (e.g. canonical labeling) refined over the last three decades. In addition, GI-Ext also solves certain GM problems, e.g. it detects important isomorphic structures induced in non-isomorphic graphs.

Twins of rayless graphs

Two non-isomorphic graphs are twins if each is isomorphic to a subgraph of the other. We prove that a rayless graph has either infinitely many twins or none.

On the spectral characterization of the union of complete multipartite graph and some isolated vertices

A graph is said to be determined by its adjacency spectrum (DS for short) if there is no other non-isomorphic graph with the same spectrum. In this paper, we focus our attention on the spectral characterization of the union of complete multipartite graph and some isolated vertices, and all its cospectral graphs are obtained. By the results, some complete multipartite graphs determined by their adjacency spectrum are also given. This extends several previous results of spectral characterization related to the complete multipartite graphs.

Pentavalent One-regular Graphs of Square-free Order

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. It is shown in this paper that a pentavalent one-regular graph of order n exists if and only if n = 2 · 5tp1p2 … ps ≥ 62, where t ≤ 1, s ≥ 1, and pi's are distinct primes such that 5|(pi-1). For such an integer n, there are exactly 4s-1 non-isomorphic pentavalent one-regular graphs of order n, which are Cayley graphs on dihedral groups constructed by Kwak et al. This work is a continuation of the classification of cubic one-regular graphs of order twice a square-free integer given by Zhou and Feng.

The lattice of integer flows of a regular matroid

For a finite multigraph G, let Λ ( G ) denote the lattice of integer flows of G – this is a finitely generated free abelian group with an integer-valued positive definite bilinear form. Bacher, de la Harpe, and Nagnibeda show that if G and H are 2-isomorphic graphs then Λ ( G ) and Λ ( H ) are isometric, and remark that they were unable to find a pair of nonisomorphic 3-connected graphs for which the corresponding lattices are isometric. We explain this by examining the lattice Λ ( M ) of integer flows of any regular matroid M . Let M • be the minor of M obtained by contracting all co-loops. We show that Λ ( M ) and Λ ( N ) are isometric if and only if M • and N • are isomorphic.

Spectra of symmetric powers of graphs and the Weisfeiler–Lehman refinements

The k-th power of an n-vertex graph X is the iterated cartesian product of X with itself. The k-th symmetric power of X is the quotient graph of certain subgraph of its k-th power by the natural action of the symmetric group. It is natural to ask if the spectrum of the k-th power – or the spectrum of the k-th symmetric power – is a complete graph invariant for small values of k, for example, for k = O ( 1 ) or k = O ( log n ) . In this paper, we answer this question in the negative: we prove that if the well-known 2 k-dimensional Weisfeiler–Lehman method fails to distinguish two given graphs, then their k-th powers – and their k-th symmetric powers – are cospectral. As it is well known, there are pairs of non-isomorphic n-vertex graphs which are not distinguished by the k-dim WL method, even for k = Ω ( n ) . In particular, this shows that for each k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th (symmetric) powers.

Parastrophically uncancellable quasigroup equations

Krstic initiated the use of cubic graphs in solving quasigroup equations. Based on his work, Krapež and Živkovic proved that there is a bijective correspondence between classes of parastrophically equivalent parastrophically uncancellable generalized quadratic functional equations on quasigroups and three-connected cubic (multi)graphs. We use the list of such graphs given in the literature to verify existing results on equations with three, four and five variables and to prove new results for equations with six variables. We start with 14 nonisomorphic graphs with ten vertices, choose a set of 14 representative parastrophically nonequivalent equations and give their general solutions. A case of equations with seven and more variables is briefly discussed. The problem of Sokhats’kyi concerning a property which distinguishes visually two parastrophically nonequivalent equations with four variables is solved.

Two-particle quantum walks applied to the graph isomorphism problem

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 (SRGs). We study the Green's functions that characterize the dynamical evolution single-particle and two-particle quantum walks on pairs of nonisomorphic SRGs and show that interacting particles can distinguish nonisomorphic 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 nonisomorphic SRGs. (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 nonisomorphic 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 nonisomorphic 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.

Some results on the Laplacian spectrum

A graph is called a Laplacian integral graph if the spectrum of its Laplacian matrix consists of integers, and a graph G is said to be determined by its Laplacian spectrum if there does not exist other non-isomorphic graph H such that H and G share the same Laplacian spectrum. In this paper, we obtain a sharp upper bound for the algebraic connectivity of a graph, and identify all the Laplacian integral unicyclic, bicyclic graphs. Moreover, we show that all the Laplacian integral unicyclic, bicyclic graphs are determined by their Laplacian spectra.

On graphs isomorphic to their neighbour and non-neighbour sets

The paper contains a construction of a universal countable graph, different from the Rado graph, such that for any of its vertices both the neighbourhood and the non-neighbourhood induce subgraphs isomorphic to the whole graph. This solves an open problem proposed by A. Bonato; see Problem 20 in Cameron (2003) [5]. We supply a construction of several non-isomorphic graphs with the property, and consider tournaments with an analogous property.

Starlike trees with maximum degree 4 are determined by their signless Laplacian spectra

A graph is said to be determined by its signless Laplacian spectrum if there is no other non-isomorphic graph with the same spectrum. In this paper, it is shown that each starlike tree with maximum degree 4 is determined by its signless Laplacian spectrum.

Non-Isomorphic Graphs with Cospectral Symmetric Powers

The symmetric $m$-th power of a graph is the graph whose vertices are $m$-subsets of vertices and in which two $m$-subsets are adjacent if and only if their symmetric difference is an edge of the original graph. It was conjectured that there exists a fixed $m$ such that any two graphs are isomorphic if and only if their $m$-th symmetric powers are cospectral. In this paper we show that given a positive integer $m$ there exist infinitely many pairs of non-isomorphic graphs with cospectral $m$-th symmetric powers. Our construction is based on theory of multidimensional extensions of coherent configurations.

Properties Determined by the Ihara Zeta Function of a Graph

In this paper, we show how to determine several properties of a finite graph $G$ from its Ihara zeta function $Z_{G}(u)$. If $G$ is connected and has minimal degree at least 2, we show how to calculate the number of vertices of $G$. To do so we use a result of Bass, and in the case that $G$ is nonbipartite, we give an elementary proof of Bass' result. We further show how to determine whether $G$ is regular, and if so, its regularity and spectrum. On the other hand, we extend work of Czarneski to give several infinite families of pairs of non-isomorphic non-regular graphs with the same Ihara zeta function. These examples demonstrate that several properties of graphs, including vertex and component numbers, are not determined by the Ihara zeta function. We end with Hashimoto's edge matrix T. We show that any graph $G$ with no isolated vertices can be recovered from its $T$ matrix. Since graphs with the same Ihara zeta function are exactly those with isospectral T matrices, this relates again to the question of what information about $G$ can be recovered from its Ihara zeta function.

On a signless Laplacian spectral characterization of [formula omitted]-shape trees

Let M be an associated matrix of a graph G (the adjacency, Laplacian and signless Laplacian matrix). Two graphs are said to be cospectral with respect to M if they have the same M spectrum. A graph is said to be determined by M spectrum if there is no other non-isomorphic graph with the same spectrum with respect to M . It is shown that T -shape trees are determined by their Laplacian spectra. Moreover among them those are determined by their adjacency spectra are characterized. In this paper, we identify graphs which are cospectral to a given T -shape tree with respect to the signless Laplacian matrix. Subsequently, T -shape trees which are determined by their signless Laplacian spectra are identified.

Complete triangulations of a given order generated from a multitude of nonisomorphic cubic graphs by current assignments

AbstractIt is known that for all sufficiently large s, there are at least \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$(\frac{5}{3})$\end{document}2s nonequivalent graceful labellings of the path on 2s + 1 vertices. Using this result, we construct exponentially many index one current graphs with current group ℤ12s + 7 such that many of the current graphs have different underlying graphs. The constructed current graphs for all sufficiently large sgenerate at least 30s nonisomorphic triangular embeddings of K12s + 7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 324–334, 2009

Reconstructing Extended Perfect Binary One-Error-Correcting Codes From Their Minimum Distance Graphs

The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.

Intersection graphs of ideals of rings

In this paper, we consider the intersection graph G ( R ) of nontrivial left ideals of a ring R . We characterize the rings R for which the graph G ( R ) is connected and obtain several necessary and sufficient conditions on a ring R such that G ( R ) is complete. For a commutative ring R with identity, we show that G ( R ) is complete if and only if G ( R [ x ] ) is also so. In particular, we determine the values of n for which G ( Z n ) is connected, complete, bipartite, planar or has a cycle. Next, we characterize finite graphs which arise as the intersection graphs of Z n and determine the set of all non-isomorphic graphs of Z n for a given number of vertices. We also determine the values of n for which the graph of Z n is Eulerian and Hamiltonian.

6-Critical Graphs on the Klein Bottle

We provide a complete list of 6-critical graphs that can be embedded on the Klein bottle settling a problem of Thomassen [J. Combin. Theory Ser. B, 70 (1997), pp. 67–100, Problem 3]. The list consists of nine nonisomorphic graphs which have altogether 18 nonisomorphic 2-cell embeddings and one embedding that is not 2-cell.