Non-isomorphic Pairs Research Articles

Hearing shapes viap-adic Laplacians

For a finite graph, a spectral curve is constructed as the zero set of a two-variate polynomial with integer coefficients coming from p-adic diffusion on the graph. It is shown that certain spectral curves can distinguish non-isomorphic pairs of isospectral graphs, and can even reconstruct the graph. This allows the graph reconstruction from the spectrum of the associated p-adic Laplacian operator. As an application to p-adic geometry, it is shown that the reduction graph of a Mumford curve and the product reduction graph of a p-adic analytic torus can be recovered from the spectrum of such operators.

Entanglement entropy in the ground state of supersymmetric fermion lattice models

In this paper, we study the entanglement entropy for strongly correlated spinless fermions in the ground state of a supersymmetric Hamiltonian on diverse graphs. Then, we use the entanglement entropy to study the graph isomorphism (GI) problem on some non-isomorphic pairs of graphs to distinguish them from each other. The GI problem is considered on pairs of non-isomorphic strongly regular graphs (SRG) and some cases of regular graphs. The SRGs are well-known for being highly symmetric, so that they rarely can be detected by classical and quantum algorithms. However, we show that most of the pairs of non-isomorphic SRGs can be distinguished by utilizing the supersymmetric entanglement entropy. Also, detecting the regular non-isomorphic graphs is often difficult. The obtained result illustrates that the entanglement entropy of the single-particle Hamiltonian cannot detect non-isomorphic pairs of regular graphs, but the entanglement entropy of the two-particle Hamiltonian can.

Torsion points of small order on hyperelliptic curves

Let $$\mathscr {C}$$ be a hyperelliptic curve of genus $$g>1$$ over an algebraically closed field K of characteristic zero and $$\mathscr {O}$$ one of the $$(2g{+}2)$$ Weierstrass points in $$\mathscr {C}(K)$$ . Let J be the Jacobian of $$\mathscr {C}$$ , which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of $$\mathscr {C}$$ into J that sends $$\mathscr {O}$$ to the zero of the group law on J. This embedding allows us to identify $$\mathscr {C}(K)$$ with a certain subset of the commutative group J(K). A special case of the famous theorem of Raynaud (Manin–Mumford conjecture) asserts that the set of torsion points in $$\mathscr {C}(K)$$ is finite. It is well known that the points of order 2 in $$\mathscr {C}(K)$$ are exactly the “remaining” $$(2g{+}1)$$ Weierstrass points. One of the authors (Zarhin in Izv Math 83:501–520, 2019) proved that there are no torsion points of order n in $$\mathscr {C}(K)$$ if $$3\leqslant n\leqslant 2g$$ . So, it is natural to study torsion points of order $$2g+1$$ (notice that the number of such points in $$\mathscr {C}(K)$$ is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually non-isomorphic pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least four points of order $$2g+1$$ . In the present paper we prove that (for a given g) there are at most finitely many (up to an isomorphism) pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least six points of order $$2g+1$$ .

Seurat games on Stockmeyer graphs

AbstractWe define a family of vertex colouring games played over a pair of graphs or digraphs by players and . These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number such that always has a winning strategy in the game with colours whenever . This is related to the reconstruction conjecture for graphs and the degree‐associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with for graphs, and the same is true for the degree‐associated reconstruction conjecture and our conjecture for digraphs. We show (for any ) that the 2‐colour game can distinguish certain nonisomorphic pairs of graphs that cannot be distinguished by the ‐dimensional Weisfeiler–Leman algorithm. We also show that the 2‐colour game can distinguish the nonisomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture.

Reconstruction of Rooted Directed Trees

Let T be a rooted directed tree on n vertices, rooted at v. The rooted subtree frequency vector (RSTF-vector) of T with root v, denoted by rstf(T, v) is a vector of length n whose entry at position k is the number of subtrees of T that contain v and have exactly k vertices. In this paper we present an algorithm for reconstructing rooted directed trees from their rooted subtree frequencies (up to isomorphism). We show that there are examples of nonisomorphic pairs of rooted directed trees that are RSTF-equivalent, s.t. they share the same rooted subtree frequency vectors. We have found all such pairs (groups) for small sizes by using exhaustive computer search. We show that infinitely many nonisomorphic RSTF-equivalent pairs of trees exist by constructing infinite families of examples.

Topological Sensitivity in the Recognition of Disoriented Figures.

A previous study by the author found that discrimination latencies for figure pairs with the same topological structure (isomorphic pairs) were longer than for pairs with different topological structures (nonisomorphic pairs). These results suggest that topological sensitivity occurs during figure recognition. However, sameness was judged in terms of both shape and orientation. Using this criterion, faster discrimination of nonisomorphic pairs may have arisen from the detection of differences in the corresponding locations of the paired figures, which is not a topological property. The current study examined whether topological sensitivity occurs even when identity judgments are based on the sameness of shapes, irrespective of their orientation, where the sameness of location is not ensured. The current results suggested the involvement of topological sensitivity, indicating that processing of structural properties (invariant features) of a figure may be prioritized over processing of superficial features, such as location, length, and angles, in figure recognition.

Entanglement entropy in the spinless free fermion model and its application to the graph isomorphism problem

In this paper, the entanglement entropy is investigated in the ground state of a spinless free fermion Hamiltonian where its hopping matrix is given by the adjacency matrix of the graph. The bipartite entanglement entropy is calculated from the eigenvalues of the correlation matrix. The entanglement entropy and mutual information are calculated for some scalable graphs including a complete graph and three sets of strongly regular graphs (SRGs). In addition, the volume law scaling has been shown for these graphs. Then, we use the entanglement entropy as a tool for studying the graph isomorphism problem in some non-isomorphic pairs of graphs in order to distinguish them from each other. Two graphs are isomorphic when they are related to each other by relabeling the graph vertices. We study the graph isomorphism in non-isomorphic SRGs, distance-regular graphs and pseudo distance-regular graphs. Most pairs of non-isomorphic SRGs can be distinguished from each other by using fermionic entanglement entropy.

Entanglement entropy of free fermions on directed graphs

In this paper, we study the general spinless quadratic fermion Hamiltonian with interaction matrices given by the symmetric and antisymmetric parts of the adjacency matrix of a directed graph. The correlation matrix and entanglement entropy are provided for the ground state of the Hamiltonian, analytically. We also show that a volume law scaling holds for some scalable sets of nonsymmetric association-scheme graphs. The scaling of the entanglement entropy is then used as a tool for studying the graph isomorphism problem, in particular to distinguish some nonisomorphic pairs of directed strongly regular graphs.

QAPV: a polynomial invariant for graph isomorphism testing

To each instance of the Quadratic Assignment Problem (QAP) a relaxed instance can be associated. Both variances of their solution values can be calculated in polynomial time. The graph isomorphism problem (GIP) can be modeled as a QAP, associating its pair of data matrices with a pair of graphs of the same order and size. We look for invariant edge weight functions for the graphs composing the instances in order to try to find quantitative differences between variances that could be associated with the absence of isomorphism. This technique is sensitive enough to show the effect of a single edge exchange between two regular graphs of up to 3,000 vertices and 300,000 edges with degrees up to 200. Planar graph pairs from a dense family up to 300,000 vertices were also discriminated. We conjecture the existence of functions able to discriminate non-isomorphic pairs for every instance of the problem.

On the Generality of the Topological Theory of Visual Shape Perception

This study used a series of six closely related experiments to examine whether individuals use topological structures to discriminate figures. Strict control was exerted over the selection of stimuli, which were a specific type of randomly generated lined figures that can be classified using isomorphic sets defined by graph theory. Any two figures within an isomorphic set possessed the same topological structure. The experiments described here used a same/different discrimination task with simultaneously presented pairs of figures: (a) identical pairs (Id pairs), in which each pair of figures had the same topological and superficial properties; (b) nonidentical and isomorphic pairs (Iso pairs), in which each pair had the same topological but different superficial properties; and (c) nonidentical and nonisomorphic pairs (Noniso pairs), in which each pair had different topological properties. Within these experiments I varied the conditions related to the intersecting line segments, presentation of points defining each figure, figure complexity, stimulus aspect ratios, and the parity of the total line-segment lengths between the figures in each pair. These variations showed that the latencies for making accurate discriminations were shorter for Noniso pairs than for Iso pairs, suggesting that individuals are sensitive to topology when distinguishing figures.

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.

Elementary equivalence of profinite groups

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different. If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the classification of the finite simple groups. Our result does not hold if the profinite groups are not finitely generated. We give concrete examples of non-isomorphic profinite groups which are elementarily equivalent.

A pair of nonisomorphic $\equiv_{\infty \lambda}$ models of power $\lambda$ for $\lambda$ singular with $\lambda_\omega=\lambda$.

On considere L∞λ la logique infinitaire formee en permettant des conjonctions arbitraires de formules et soit des quantifications existentielles soit des quantifications universelles sur des ensembles de plus de λ variables. On demontre que si λω=λ, alors il y a des structures M, N, avec ∥M∥=∥N∥=λ telles que M≡∞λN, mais M ≅ N