Walks In Graphs Research Articles

Setting port numbers for fast graph exploration

We consider the problem of periodic graph exploration by a finite automaton in which an automaton with a constant number of states has to explore all unknown anonymous graphs of arbitrary size and arbitrary maximum degree. In anonymous graphs, nodes are not labeled but edges are labeled in a local manner (called local orientation) so that the automaton is able to distinguish them. Precisely, the edges incident to a node v are given port numbers from 1 to d v , where d v is the degree of v . Periodic graph exploration means visiting every node infinitely often. We are interested in the length of the period, i.e., the maximum number of edge traversals between two consecutive visits of any node by the automaton in the same state and entering the node by the same port. This problem is unsolvable if local orientations are set arbitrarily. Given this impossibility result, we address the following problem: what is the minimum function π ( n ) such that there exists an algorithm for setting the local orientation, and a finite automaton using it, such that the automaton explores all graphs of size n within the period π ( n ) ? The best result so far is the upper bound π ( n ) ≤ 10 n , by Dobrev et al. [S. Dobrev, J. Jansson, K. Sadakane, W.-K. Sung, Finding short right-hand-on-the-wall walks in graphs, in: 12th Colloquium on Structural Information and Communication Complexity, SIROCCO, in: LNCS, vol. 3499, 2005, pp. 127–139], using an automaton with no memory (i.e. only one state). In this paper we prove a better upper bound π ( n ) ≤ 4 n . Our automaton uses three states but performs periodic exploration independently of its starting position and initial state.

Graph Laplacians and their Convergence on Random Neighborhood Graphs

Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a non-uniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted Laplace-Beltrami operator.

Fast periodic graph exploration with constant memory

We consider the problem of periodic exploration of all nodes in undirected graphs by using a finite state automaton called later a robot. The robot, using a constant number of states (memory bits), must be able to explore any unknown anonymous graph. The nodes in the graph are neither labelled nor coloured. However, while visiting a node v the robot can distinguish between edges incident to it. The edges are ordered and labelled by consecutive integers 1 , … , d ( v ) called port numbers, where d ( v ) is the degree of v. Periodic graph exploration requires that the automaton has to visit every node infinitely many times in a periodic manner. In this paper, we are interested in minimisation of the length of the exploration period. In other words, we want to minimise the maximum number of edge traversals performed by the robot between two consecutive visits of a generic node, in the same state and entering the node by the same port. Note that the problem is unsolvable if the local port numbers are set arbitrarily, see [L. Budach, Automata and labyrinths, Math. Nachr. 86 (1978) 195–282]. In this context, we are looking for the minimum function π ( n ) , such that, there exists an efficient deterministic algorithm for setting the local port numbers allowing the robot to explore all graphs of size n along a traversal route with the period π ( n ) . Dobrev et al. proved in [S. Dobrev, J. Jansson, K. Sadakane, W.-K. Sung, Finding short right-hand-on-the-wall walks in graphs, in: Proc. 12th Colloquium on Structural Information and Communication Complexity, SIROCCO 2005, in: Lecture Notes in Comput. Sci., vol. 3499, Springer, Berlin, 2005, pp. 127–139] that for oblivious robots π ( n ) ⩽ 10 n . Recently Ilcinkas proposed another port labelling algorithm for robots equipped with two extra memory bits, see [D. Ilcinkas, Setting port numbers for fast graph exploration, in: Proc. 13th Colloquium on Structural Information and Communication Complexity, SIROCCO 2006, in: Lecture Notes in Comput. Sci., vol. 4056, Springer, Berlin, 2006, pp. 59–69], where the exploration period π ( n ) ⩽ 4 n − 2 . In the same paper, it is conjectured that the bound 4 n − O ( 1 ) is tight even if the use of larger memory is allowed. In this paper, we disprove this conjecture presenting an efficient deterministic algorithm arranging the port numbers, such that, the robot equipped with a constant number of bits is able to complete the traversal period in π ( n ) < 3.75 n − 2 steps hence decreasing the existing upper bound. This reduces the gap with the lower bound of π ( n ) ⩾ 2 n − 2 holding for any robot.

Enabling access-privacy for random walk based data analysis applications

Random walk graph and Markov chain based models are used heavily in many data and system analysis domains, including web, bioinformatics, and queuing. These models enable the description and analysis of various behaviors of stochastic systems. If the system being modeled has certain properties, such as if it is irreducible and aperiodic, close form formulations corresponding to its stationary behavior can be used to analyze its behavior. However, if the system does not have these properties or if the user is not interested in the stationary behavior, then an iterative approach needs to be used to determine potential outcomes based on the initial probability distribution inputs to the model. In this paper, we focus on access-privacy enabled outsourced Markov chain based data analysis applications, where a non-trusted service provider takes (hidden) user queries that are described in terms of initial state distributions, and evaluates them iteratively in an oblivious manner. We show that this iterative process can leak information regarding the possible values of the hidden input if the server has a priori knowledge about the underlying Markovian process. Hence as opposed to simple obfuscation mechanisms, we develop an algorithm based on methodical addition of extra states, which guarantees unbounded feasible regions for the inputs, thus preventing a malicious host from having an informed guess regarding the inputs. In particular, we show that if the underlying transition matrix is diagonalizable then we can compute the exact number of states needed for access-privacy, while if the matrix is non-diagonalizable, then only a lower-bound can be computed.

RECURRENCE FOR PERSISTENT RANDOM WALKS IN TWO DIMENSIONS

We discuss the question of recurrence for persistent, or Newtonian, random walks in ℤ2, i.e. random walks whose transition probabilities depend both on the walker's position and incoming direction. We use results by Tóth and Schmidt–Conze to prove recurrence for a large class of such processes, including all "invertible" walks in elliptic random environments. Furthermore, rewriting our Newtonian walks as ordinary random walks in a suitable graph, we gain a better idea of the geometric features of the problem, and obtain further examples of recurrence.

Walks and regular integral graphs

We establish a useful correspondence between the closed walks in regular graphs and the walks in infinite regular trees, which, after counting the walks of a given length between vertices at a given distance in an infinite regular tree, provides a lower bound on the number of closed walks in regular graphs. This lower bound is then applied to reduce the number of the feasible spectra of the 4-regular bipartite integral graphs by more than a half. Next, we give the details of the exhaustive computer search on all 4-regular bipartite graphs with up to 24 vertices, which yields a total of 47 integral graphs.

On the number of closed walks in vertex-transitive graphs

The results of Širáň and the first author [A construction of vertex-transitive non-Cayley graphs, Australas. J. Combin. 10 (1994) 105–114; More constructions of vertex-transitive non-Cayley graphs based on counting closed walks, Australas. J. Combin. 14 (1996) 121–132] are generalized, and new formulas for the number of closed walks of length p r or pq, where p and q are primes, valid for all vertex-transitive graphs are found. Based on these formulas, several simple tests for vertex-transitivity are presented, as well as lower bounds on the orders of the smallest vertex- and arc-transitive groups of automorphisms for vertex-transitive graphs of given valence.

Mitochondrial-encoded membrane protein transcripts are pyrimidine-rich while soluble protein transcripts and ribosomal RNA are purine-rich

BackgroundEukaryotic organisms contain mitochondria, organelles capable of producing large amounts of ATP by oxidative phosphorylation. Each cell contains many mitochondria with many copies of mitochondrial DNA in each organelle. The mitochondrial DNA encodes a small but functionally critical portion of the oxidative phosphorylation machinery, a few other species-specific proteins, and the rRNA and tRNA used for the translation of these transcripts. Because the microenvironment of the mitochondrion is unique, mitochondrial genes may be subject to different selectional pressures than those affecting nuclear genes.ResultsFrom an analysis of the mitochondrial genomes of a wide range of eukaryotic species we show that there are three simple rules for the pyrimidine and purine abundances in mitochondrial DNA transcripts. Mitochondrial membrane protein transcripts are pyrimidine rich, rRNA transcripts are purine-rich and the soluble protein transcripts are purine-rich. The transitions between pyrimidine and purine-rich regions of the genomes are rapid and are easily visible on a pyrimidine-purine walk graph. These rules are followed, with few exceptions, independent of which strand encodes the gene. Despite the robustness of these rules across a diverse set of species, the magnitude of the differences between the pyrimidine and purine content is fairly small. Typically, the mitochondrial membrane protein transcripts have a pyrimidine richness of 56%, the rRNA transcripts are 55% purine, and the soluble protein transcripts are only 53% purine.ConclusionThe pyrimidine richness of mitochondrial-encoded membrane protein transcripts is partly driven by U nucleotides in the second codon position in all species, which yields hydrophobic amino acids. The purine-richness of soluble protein transcripts is mainly driven by A nucleotides in the first codon position. The purine-richness of rRNA is also due to an abundance of A nucleotides. Possible mechanisms as to how these trends are maintained in mtDNA genomes of such diverse ancestry, size and variability of A-T richness are discussed.

Straight-ahead walks in Eulerian graphs

A straight-ahead walk in an embedded Eulerian graph G always passes from an edge to the opposite edge in the rotation at the same vertex. A straight-ahead walk is called Eulerian if all the edges of the embedded graph G are traversed in this way starting from an arbitrary edge. An embedding that contains an Eulerian straight-ahead walk is called an Eulerian embedding. In this article, we characterize some properties of Eulerian embeddings of graphs and of embeddings of graphs such that the corresponding medial graph is Eulerian embedded. We prove that in the case of 4-valent planar graphs, the number of straight-ahead walks does not depend on the actual embedding in the plane. Finally, we show that the minimal genus over Eulerian embeddings of a graph can be quite close to the minimal genus over all embeddings.

The number of walks in a graph

The aim of this note is to call attention to a simple regularity regarding the number of walks in a finite graph G. Let W k denote the number of walks of length k(≥ 0) in G. Then W a+ b 2 ≤ W 2 a W 2 b holds for all a, b ϵ N 0 while equality holds exclusively either 1. (I) for all a, b ϵ N 0 (in case G is a regular graph), or 2. (II) for all a, b ϵ N , or 3. (III) for all a, b ϵ N 0 of equal parity (provided G is connected, this holds if and only if G is nonregular, yet semiregular graph), or 4. (IV) for all a, b ϵ N of equal parity, or 5. (V) just for a = b only. We show that all of these five cases can actually occur and discuss the resulting classification graphs in exactly five classes.

The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs

The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs

Regression Testing of Database Applications

Database applications features such as Structured Query Language or SQL, exception programming, integrity constraints, and table triggers pose difficulties for maintenance activities; especially for regression testing that follows modifications to database applications. In this work, we address these difficulties and propose a two-phase regression testing methodology. In phase 1, we explore control flow and data flow analysis issues of database applications. Then, we propose an impact analysis technique that is based on dependencies that exist among the components of database applications. This analysis leads to selecting test cases from the initial test suite for regression testing the modified application. In phase 2, further reduction in the regression test cases is performed by using reduction algorithms. We present two such algorithms. The Graph Walk algorithm walks through the control flow graph of database modules and selects a safe set of test cases to retest. The Call Graph Firewall algorithm uses a firewall for the inter-procedural level. Finally, a maintenance environment for database applications is described. Our experience with this regression testing methodology shows that the impact analysis technique is adequate for selecting regression tests and that phase 2 techniques can be used for further reduction in the number of theses tests.

On walks in molecular graphs.

Walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry. These are based on the fact that the (i, j)-entry of the kth power of the adjacency matrix is equal to the number of walks starting at vertex i, ending at vertex j, and having length k. In recent papers (refs 13, 18, 19) the numbers of all walks of length k, called molecular walk counts, mwc(k), and their sum from k = 1 to k = n - 1, called total walk count, twc, were proposed as quantities suitable for QSPR studies and capable of measuring the complexity of organic molecules. We now establish a few general properties of mwc's and twc among which are the linear dependence between the mwc's and linear correlations between the mwc's and twc, the spectral decomposition of mwc's, and various connections between the walk counts and the eigenvalues and eigenvectors of the molecular graph. We also characterize the graphs possessing minimal and maximal walk counts.

Disjoint paths, planarizing cycles, and spanning walks

We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected triangulation of a surface with sufficiently large representativity is hamiltonian, thus verifying a conjecture of Thomassen. We also obtain results about spanning walks in graphs embedded in a surface with large representativity.

On the second eigenvalue and random walks in randomd-regular graphs

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, λ2, of (the adjacency matrix of) a randomd-regular graph,G. In order to do so, we study the probability that a random walk on a random graph returns to its originating vertex at thek-th step, for various values ofk. Our main theorem about eigenvalues is that $$E|\lambda _2 (G)|^m \leqslant \left( {2\sqrt {2d - 1} \left( {1 + \frac{{\log d}}{{\sqrt {2d} }} + 0\left( {\frac{1}{{\sqrt d }}} \right)} \right) + 0\left( {\frac{{d^{3/2} \log \log n}}{{\log n}}} \right)} \right)^m $$ for any\(m \leqslant 2\left\lfloor {log n\left\lfloor {\sqrt {2d - } 1/2} \right\rfloor /\log d} \right\rfloor \), where E denotes the expected value over a certain probability space of 2d-regular graphs. It follows, for example, that for fixedd the second eigenvalue's magnitude is no more than\(2\sqrt {2d - 1} + 2\log d + C'\) with probability 1−n −C for constantsC andC′ for sufficiently largen.

On a resonance energy model based on expansion in terms of acyclic moments: Exact results

The new concept of the resonance energy in conjugated hydrocarbons introduced by Jiang Y, Zhang H (1989) Theor Chim Acta 75:279 is further developed. This model is based on expansion of the π-electron energy in terms of moments which are also equal to numbers of closed walks in a molecular graph. The reference system is established by counting only acyclic walks, i.e. those tracing only on acyclic subgraphs. Because acyclic walks could be counted only up to some finite length, the energy of the reference system has been evaluated by truncating higher terms in the expansion. In this paper a finite expression for the energy of the same reference system is derived, thus allowing its exact evaluation. The exact values differ significantly from the truncated ones. This difference, as well as the discrepancy between exact results and chemical experience, are discussed.

Matchings and walks in graphs

AbstractThe matching polynomial α(G, x) of a graph G is a form of the generating function for the number of sets of k independent edges of G. in this paper we show that if G is a graph with vertex v then there is a tree T with vertex w such that \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{\alpha (G\backslash v, x)}}{{\alpha (G, x)}} = \frac{{\alpha (T\backslash w, x)}}{{\alpha (T, x)}}. $\end{document}This result has a number of consequences. Here we use it to prove that α(G\v, 1/x)/xα(G, 1/x) is the generating function for a certain class of walks in G. As an application of these results we then establish some new properties of α(G, x).

On Hamiltonian Walks in Graphs

A Hamiltonian walk in a graph G is a closed walk of minimum length which contains every point of G. An Eulerian walk in a graph G is a closed walk of minimum length which contains every line of G. In this paper we establish several relationships between Hamiltonian and Eulerian walks. We also derive a number of bounds on the length of a Hamiltonian walk.

Eulerian Walks in Graphs

The general problem of finding the shortest edge covering walks in an arbitrary undirected graph is investigated. An exact combinatoric expression is obtained for the length of such a walk, and the graph theoretic properties of these walks are studied. Explicit solutions are exhibited for some important classes of graphs.

On minimum walks in graphs

AbstractIn this paper homomorphisms are used to define four new parameters of a graph, which have to do with minimum walks in graphs satisfying certain conditions. Several bounds and exact equations are obtained for two of these parameters; a third parameter is seen to be closely related to the Traveling Salesman Problem and the question of whether a given graph is Hamiltonian.