Automorphisms Of Graph Research Articles

A Graph Symmetrization Bound on Channel Information Leakage Under Blowfish Privacy

Blowfish privacy is a recent generalisation of differential privacy that enables improved utility while maintaining privacy policies with semantic guarantees, a factor that has driven the popularity of differential privacy in computer science. This paper relates Blowfish privacy to an important measure of privacy loss of information channels from the communications theory community: min-entropy leakage. Symmetry in an input data neighbouring relation is central to known connections between differential privacy and min-entropy leakage. But while differential privacy exhibits strong symmetry, Blowfish neighbouring relations correspond to arbitrary simple graphs owing to the framework’s flexible privacy policies. To bound the min-entropy leakage of Blowfish-private mechanisms we organise our analysis over symmetrical partitions corresponding to orbits of graph automorphism groups. A construction meeting our bound with asymptotic equality demonstrates tightness.

A note on some distance-based topological indices of circulant network Cn (1, a)

Let Г be a simple connected graph/network with vertex set V(Г) and edge set E(Г). A topological index is a real number associated to Г that characterizes its topology and is invariant under graph automorphism. If υi and υj are vertices in V(Г), the distance between them denoted by d Г (υi , υj ) refers to the length of the shortest path that connects υi , and υj . A topological index is said to be distance-based if its computation involves distance between vertices. Recently, the exact value of some distance-based topological indices namely Wiener, hyper-Wiener, and Schultz molecular topological index of the circulant network Cn (1, a) for a = 2, 3, 4, and 5 were computed. In this paper, we use the breadth-first search method to compute for some distance-based topological indices of the circulant network Cn (1, a) where a = 6 and . We also provide a general formula for the computation of the Wiener, Schultz, and Gutman index of the circulant network Cn (1, a) where .

On the Number of Fixed Points of Automorphisms of Vertex-Transitive Graphs

On the Number of Fixed Points of Automorphisms of Vertex-Transitive Graphs

Bipolar Picture Fuzzy Graphs with Application

In this manuscript, we introduce and discuss the term bipolar picture fuzzy graphs along with some of its fundamental characteristics and applications. We also initiate the concepts of complete bipolar picture fuzzy graphs and strong bipolar picture fuzzy graphs. Firstly, we apply different types of operations to bipolar picture fuzzy graphs and then we introduce various products of bipolar picture fuzzy graphs. Several other terms such as order and size, path, neighbourhood degrees, busy values of vertices and edges of bipolar picture fuzzy graphs are also discussed. These terminologies also lay the foundations for the discussion about the regular bipolar picture fuzzy graphs. Moreover, we also discuss isomorphisms, weak and co-weak isomorphisms and automorphisms of bipolar picture fuzzy graphs. Finally, at the base of bipolar picture fuzzy graph we present the construction of a bipolar picture fuzzy acquaintanceship graph, which would be an important tool to measure the symmetry or asymmetry of acquaintanceship levels of social networks, computer networks etc.

Free groups as end homogeneity groups of 3-manifolds

For every finitely generated free group F , we construct an irreducible open 3-manifold M_F whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of M_F isomorphic to F . The end homogeneity group is the group of all self-homeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends an earlier result that constructs, for each finitely generated abelian group G , an irreducible open 3-manifold M_G with end homogeneity group G . The method used in the proof of our main result also shows that if G is a group with a Cayley graph in \mathbb{R}^3 such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold M_G with end homogeneity group G .

Generalized Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators

Two distinct projections of finite rank m are adjacent if their difference is an operator of rank two or, equivalently, the intersection of their images is (m−1)-dimensional. We extend this adjacency relation on other conjugacy classes of finite-rank self-adjoint operators which leads to a natural generalization of Grassmann graphs. Let C be a conjugacy class formed by finite-rank self-adjoint operators with eigenspaces of dimension greater than 1. Under the assumption that operators from C have at least three eigenvalues we prove that every automorphism of the corresponding generalized Grassmann graph is the composition of an automorphism induced by a unitary or anti-unitary operator and the automorphism obtained from a permutation of eigenspaces with the same dimensions. The case when the operators from C have two eigenvalues only is covered by classical Chow's theorem which says that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality.

Automorphisms of linear functional graphs over vector spaces

Let be a finite field with q elements, a positive integer, a n-dimensional vector space over and the set of all linear functionals from to . Let and . The linear functional graph of dented by , is an undirected bipartite graph, whose vertex set V is partitioned into two sets as and two vertices and are adjacent if and only if f sends v to the zero element of (i.e. ). In this paper, the structure of all automorphisms of this graph is characterized and formalized. Also the cardinal number of automorphisms group for this graph is determined.

Revisiting Parameter Estimation in Biological Networks: Influence of Symmetries

Graph models often give us a deeper understanding of real-world networks. In the case of biological networks they help in predicting the evolution and history of biomolecule interactions, provided we map properly real networks into the corresponding graph models. In this paper, we show that for biological graph models many of the existing parameter estimation techniques overlook the critical property of graph symmetry (also known formally as graph automorphisms), thus the estimated parameters give statistically insignificant results concerning the observed network. To demonstrate it and to develop accurate estimation procedures, we focus on the biologically inspired duplication-divergence model, and the up-to-date data of protein-protein interactions of seven species including human and yeast. Using exact recurrence relations of some prominent graph statistics, we devise a parameter estimation technique that provides the right order of symmetries and uses phylogenetically old proteins as the choice of seed graph nodes. We also find that our results are consistent with the ones obtained from maximum likelihood estimation (MLE). However, the MLE approach is significantly slower than our methods in practice.

Bruhat order on the Fixed-point subgroup by a Coxeter graph automorphism

Bruhat order on the Fixed-point subgroup by a Coxeter graph automorphism

Computational Complexity of Computing Symmetries in Finite-Domain Planning

Symmetry-based pruning is a powerful method for reducing the search effort in finitedomain planning. This method is based on exploiting an automorphism group connected to the ground description of the planning task { these automorphisms are known as structural symmetries. In particular, we are interested in the StructSym problem where the generators of this group are to be computed. It has been observed in practice that the StructSym problem is surprisingly easy to solve. We explain this phenomenon by showing that StructSym is GI-complete, i.e., the graph isomorphism problem is polynomial-time equivalent to it and, consequently, solvable in quasi-polynomial time. This implies that it is solvable substantially faster than most computationally hard problems encountered in AI. We accompany this result by identifying natural restrictions of the planning task and its causal graph that ensure that StructSym can be solved in polynomial time. Given that the StructSym problem is GI-complete and thus solvable quite efficiently, it is interesting to analyse if other symmetries (than those that are encompassed by the StructSym problem) can be computed and/or analysed efficiently, too. To this end, we present a highly negative result: checking whether there exists an automorphism of the state transition graph that maps one state s into another state t is a PSPACE-hard problem and, consequently, at least as hard as the planning problem itself.

On some 2-edge-connected homogenous graphs and Cayley graphs with two removable vertices

A graph $$\Gamma$$ is called homogenous if the group of automorphisms of $$\Gamma$$ acts transitively on the set of vertices of $$\Gamma$$ . We will study the following properties for homogenous graphs: Removability: a vertex p is removable if the graph obtained by removing it is isomorphic to the original. Property (R): for every edge x there exists an edge y such that removing x and y disconnects the graph. We will study also Cayley graphs of groups of automorphisms of graphs with the property (R) and related concepts.

Symmetric graphs of valency 4 having a quasi-semiregular automorphism

Feng et al. (2019) characterized connected G-symmetric graphs of valency 4 having a quasi-semiregular automorphism, namely, a graph automorphism fixing a unique vertex in the vertex set of the graph and keeping the lengths of all other orbits equal, when G is soluble or the vertex stabilizer in G is not a 2-group. In this paper we prove that a connected symmetric graph with valency 4 having a quasi-semiregular automorphism is a Cayley graph on a group G with respect to S, where G is abelian of odd order and S is an orbit of a group of automorphisms of the group G.

Folding of Hitchin Systems and Crepant Resolutions

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.

Automorphisms of a Distance Regular Graph with Intersection Array $$\{21,18,12,4;1,1,6,21\}$$

Automorphisms of a Distance Regular Graph with Intersection Array $$\{21,18,12,4;1,1,6,21\}$$

Exploiting Symmetry Reduces the Cost of Training QAOA

A promising approach to the practical application of the Quantum Approximate Optimization Algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. We propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization RQAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10,000 nodes. Our approach provides an improvement for $p=1$ on $71.7\%$ of the graphs considered, with a median speedup of $4.06$, on a benchmark where $62.5\%$ of the graphs are known to be hard for automorphism solvers.

DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST

In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3,18\}\), and \(\{27,20,7;1,4,21\}\). Automorphisms of graphs with intersection arrays \(\{18,15,9;1,1,10\}\) and \(\{24,21,3;1,3,18\}\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \(\{27,20,7;1,4,21\}\) does not exist.

Automorphisms of Contact Graphs of CAT(0) Cube Complexes

Abstract We show that, under weak assumptions, the automorphism group of a $\textrm{CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen’s contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov’s theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim–Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated with Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex.

Graph automorphisms for compression

Abstract Detecting automorphisms is a natural way to identify redundant information presented in structured data. When such redundancies are detected they can be used for data compression. In this paper we explore two different classes of graphs to capture this intuitive property of automorphisms. Symmetry-compressible graphs are the first class which introduces the basic concepts but use only global symmetries for the compression. In order for this concept to be more practical, we need to use local symmetries. Thus, we extend the basic graph class with Near Symmetry compressible graphs. Furthermore, we develop two algorithms that can be used to compress practical instances and empirically evaluate them on a set of realistic graphs.

On Knowledge Discovery and Representations of Molecular Structures Using Topological Indices

Abstract The main purpose of a topological index is to encode a chemical structure by a number. A topological index is a graph invariant, which decribes the topology of the graph and remains constant under a graph automorphism. Topological indices play a wide role in the study of QSAR (quantitative structure-activity relationship) and QSPR (quantitative structure-property relationship). Topological indices are implemented to judge the bioactivity of chemical compounds. In this article, we compute the ABC (atom-bond connectivity); ABC 4 (fourth version of ABC), GA (geometric arithmetic) and GA 5 (fifth version of GA) indices of some networks sheet. These networks include: octonano window sheet; equilateral triangular tetra sheet; rectangular sheet; and rectangular tetra sheet networks.

On A Rough Cayley Graph Related to Conjugacy Classes

In this paper, we discuss concepts of lower and upper approximations edge Cayley graphs of Cayley graphs with respect to conjugacy classes. Also, we expand rough approximation to pseudo-Cayley graphs and introduce concept of lower and upper approximations vertex pseudo-Cayley graphs of them with respect to a conjugacy classes. In the following, we discuss the properties of automorphisms in the Cayley rough graphs.