Downloads & Free Reading Options - Results

We investigate Legendrian graphs in $(\R^3, \xi_{std})$. We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with $tb=-1$ and $rot=0$ if and only if it does not contain $K_4$ as a minor. We show that the pair $(tb, rot)$ does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. For the lollipop graph the pair $(tb,rot)$ determines two Legendrian classes and for the handcuff graph it determines four Legendrian classes.

We investigate Legendrian graphs in $(\R^3, \xi_{std})$. We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with $tb=-1$ and $rot=0$ if and only if it does not contain $K_4$ as a minor. We show that the pair $(tb, rot)$ does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. For the lollipop graph the pair $(tb,rot)$ determines two Legendrian classes and for the handcuff graph it determines four Legendrian classes.

This report presents a very simple algorithm for overlaping community-detection in large graphs under constraints such as the minimum and maximum number of members allowed. The algorithm is based on the simulation of random walks and measures the entropy of each random walk to detect the discovery of a community.

In the present paper, we wish to treat some relationships between toughness of a graph and the n-extendability of the graph. We will prove two results. The first says essentially that if a graph has sufficiently high toughness (and has an even number of points) then it must be n-extendable. The second result applies to graphs with toughness less than one and presents an upper bound on the value of n for which such a graph can be n-extendable. In the final section, we compare and contrast these results with the n-factor results of Enomoto, Jackson, Katerinis and A. Saito.

Planar embedding with minimal area of graphs on an integer grid is one of the major issues in VLSI. Valiant (V) gave an algorithm to construct a planar embedding for trees in linear area; he also proved that there are planar graphs that require quadratic area. We give an algorithm to embed outerplanar graphs in linear area. We extend this algorithm to work for every planar graph that has the following property: for every vertex there exists a path of length less than K to the exterior face, where K is a constant. Finally, finding a minimal embedding area is shown to be NP-complete for forests, and hence more general types of graphs. (Author)

We want to introduce a construction of spherical designs from finite graphs with the theory of crystal lattice. We start from a finite graph, and we consider standard realization of the crystal lattices as the maximal Abelian covering of the graph. Then, we take the set of vectors which form the crystal lattice. If every vector has the same norm, then we can consider a finite set on Euclidean sphere, and then we get a spherical design. In this paper, we observe the results by numerical calculations. We tried constructing vectors from various finite graphs, strongly regular graphs, distance regular graphs, and so on. We also introduce some facts and conjectures.

This paper deals with chain graphs under the alternative Andersson-Madigan-Perlman (AMP) interpretation. In particular, we present a constraint based algorithm for learning an AMP chain graph a given probability distribution is faithful to. We also show that the extension of Meek's conjecture to AMP chain graphs does not hold, which compromises the development of efficient and correct score+search learning algorithms under assumptions weaker than faithfulness.

We establish a new formula for the heat kernel on regular trees in terms of classical I-Bessel functions. Although the formula is explicit, and a proof is given through direct computation, we also provide a conceptual viewpoint using the horocyclic transform on regular trees. From periodization, we then obtain a heat kernel expression on any regular graph. From spectral theory, one has another expression for the heat kernel as an integral transform of the spectral measure. By equating these two formulas and taking a certain integral transform, we obtain several generalized versions of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. Our approach to the Ihara zeta function and determinant formula through heat kernel analysis follows a similar methodology which exists for quotients of rank one symmetric spaces.

We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this operation. The result is motivated by and applicable to the theory of gene assembly in ciliates.

For integers $k\geq 1$ and $n\geq 2k+1$ the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as vertices all $k$-element and $(n-k)$-element subsets of $[n]$ and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph $K(5,2)$ have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs $H(n,k)$. We also establish the existence of cycles that visit almost all vertices in Kneser graphs $K(n,k)$ when $n=2k+o(k)$, generalizing and improving upon previous results on this problem.

A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially 11-colourable. As a consequence, we derive that every 2-connected plane graph with maximum face-size at most 7 is cyclically 11-colourable. These two bounds are for one off from those that are proposed by the (3l+1)-Conjecture and the Cyclic Conjecture.

Background: Numerous academic disciplines commonly require students to develop skills in comprehending so-called 2x2 graphs. These graphs show a dependent variable, and two predictor variables -each with two levels (hence 2x2). Prior studies on 2x2 graph comprehension had participants either "think aloud" (Ali & Peebles, 2013; Peebles & Ali, 2015) or provide written descriptions (Shah & Freedman, 2011) about 2x2 main effects and interaction effects. Those methods privilege declarative knowledge (what can be stated). Here, we will measure perceptual learning in 2x2 graph-pattern-detection using a trial-and-error task that does not require declarative knowledge. Stimuli, Task, and Research Design: Participants will view variations on eight 2x2 graph patterns, shown as either line graphs or bar graphs. Each participant will classify 192 graphs into two initially unknown categories, Category “N” vs Category “Y”. These categories map to non-significant vs significant effects in one of three randomly assigned target factors: Factor A (left vs right height differences); Factor B (black vs white height differences); Interaction (black vs white slope differences). This psychophysical experiment has a 3x2 between-subject research design: Target Factor (“A”, “B”, “Interaction”) by Graph Type (Line versus Bar). Significance Statement: Across our Target Factor and Graph Type conditions, the mathematical information in our stimuli will remain identical. Consequently, statistically significant effects in our data will reveal biases in perceptual organization when naive participants learn to detect 2x2 graph patterns. These biases in perceptual organization have the potential to inform the applied fields of data visualization and statistics education, while furthering our knowledge about perceptual learning and ensemble perception.

A general-purpose browser for directed graphs is described. The browser provides operations to examine and edit graphs. An operation is also provided to generate a layout for graph automatically that minimizes edge crossing. Two layout algorithms were implemented. A hierarchical graph layout algorithm was found to be best for directed graphs. The graph browser also has facilities that allow it to be integrated with other applications

Consider a network of n processes each of which has a d-dimensional vector of reals as its input. Each process can communicate directly with all the processes in the system thus the communication network is a complete graph. All the communication channels are reliable and FIFO (first-in-first-out). The problem of Byzantine vector consensus (BVC) requires agreement on a d-dimensional vector that is in the convex hull of the d-dimensional input vectors at the non-faulty processes. We obtain the following results for Byzantine vector consensus in complete graphs while tolerating up to f Byzantine failures: * We prove that in a synchronous system, n or = max(3f+1; (d+1)f+1) is necessary and sufficient for achieving Byzantine vector consensus. * In an asynchronous system, it is known that exact consensus is impossible in presence of faulty processes. For an asynchronous system, we prove that n or = (d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector consensus. Our sufficiency proofs are constructive. We show sufficiency by providing explicit algorithms that solve exact BVC in synchronous systems, and approximate BVC in asynchronous systems. We also obtain tight bounds on the number of processes for achieving BVC using algorithms that are restricted to a simpler communication pattern.

Let k>0 be an integer, let H be a minor-minimal graph in the projective plane such that every homotopically non-trivial closed curve intersects H at least k times, and let G be the planar double cover of H obtained by lifting G into the universal covering space of the projective plane, the sphere. We prove that G is minor-minimal of branch-width 2k. We also exhibit examples of minor-minimal planar graphs of branch-width 6 that do not arise this way.

A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.

Cameron and Kazanidis have recently shown that rank-3 graphs are either cores or have complete cores, and they asked whether this holds for all strongly regular graphs. We prove that this is true for the point graphs and line graphs of generalized quadrangles and that when the number of points is sufficiently large, it is also true for the block graphs of Steiner systems and orthogonal arrays.

A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws lead to the "weak chromatic group" of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for chromatic functions of gain graphs. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.

In this paper, we investigate continuous-time quantum walk on star graphs. It is shown that quantum central limit theorem for a continuous-time quantum walk on star graphs for $N$-fold star power graph, which are invariant under the quantum component of adjacency matrix, converges to continuous-time quantum walk on $K_2$ graphs (Complete graph with two vertices) and the probability of observing walk tends to the uniform distribution.

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th $s$-element subset of $E(G)$ and any two vertices $v_{s,i}, v_{k,m}$ of $\mathcal G_G$ are adjacent if and only if there is at least one edge in the edge-subset corresponding to $v_{s,i}$ which is adjacent to at least one edge in the edge-subset corresponding to $v_{k,m}$ where $s,k$ are positive integers. It can be noted that the edge-set graph $\mathcal G_G$ of a graph $G$ id dependent on both the structure of $G$ as well as the number of edges $\epsilon.$ We also discuss the characteristics and properties of the edge-set graphs corresponding to certain standard graphs.

A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this paper a model is developed in which each vertex is given a random weight, and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be so as well. Furthermore, an asymptotic expression for the clustering in the graph is derived. By tuning the parameters of the model, it is possible to generate a graph with arbitrary clustering, expected degree and -- in the power law case -- tail exponent.

We determine the cells, whose existence has been announced by Ocneanu, on all the candidate nimrep graphs except $\mathcal{E}_4^{(12)}$ proposed by di Francesco and Zuber for the SU(3) modular invariants classified by Gannon. This enables the Boltzmann weights to be computed for the corresponding integrable statistical mechanical models and provide the framework for studying corresponding braided subfactors to realise all the SU(3) modular invariants as well as a framework for a new SU(3) planar algebra theory.

We consider a soluble model of large $\phi^{4}$-graphs randomly embedded in one compactified dimension; namely the large-order behaviour of finite-temperature perturbation theory for the partition function of the anharmonic oscillator. We solve the model using semi-classical methods and demonstrate the existence of a critical temperature at which the system undergoes a second-order phase transition from $D=1$ to $D=0$ behaviour. Non-trivial windings of the closed loops in a graph around the compactified time direction are interpreted as vortices. The critical point has a natural interpretation as the temperature at which these vortices condense and disorder the system. We show that the vortex density increases rapidly in the critical region indicating the breakdown of the dilute vortex gas approximation at this point. We discuss the relation of this phenomenon to the Berezinskii-Kosterlitz-Thouless transition in the $D=1$ matrix model formulated on a circle.

Let $G=(V,E)$ be an arbitrary undirected source graph to be embedded in a target graph $EM$, the extended grid with vertices on integer grid points and edges to nearest and next-nearest neighbours. We present an algorithm showing how to embed $G$ into $EM$ in both time and space $O(|V|^2)$ using the new notions of islands and bridges. An island is a connected subgraph in the target graph which is mapped from exactly one vertex in the source graph while a bridge is an edge between two islands which is mapped from exactly one edge in the source graph. This work is motivated by real industrial applications in the field of quantum computing and a need to efficiently embed source graphs in the extended grid.

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x\neq y$, alternate in $w$ if and only if $(x,y)\in E$. Halld\'{o}rsson et al.\ have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary to this result is that any 3-colorable graph is word-representable. Akrobotu et al.\ have shown that a triangulation of a grid graph is word-representable if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder graphs, namely, there are such word-representable graphs with chromatic number 4. In this paper we show that word-representability of triangulations of grid-covered cylinder graphs with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel graphs $W_5$ and $W_7$).

This study aims to explore how undergraduate students in mathematics and engineering professions make sense out of graphs representing periodic and repeated but non-periodic motions. In this study, making sense out of graphs means interpreting graphical features and describing a situation that could be represented by them. The data was collected by means of a questionnaire administered to 132 participants. Our findings indicated both students' misconceptions, as every repeated motion is periodical, and their strong willingness to assign practical meaning to mathematical entities. [For the complete proceedings, see ED597799.]

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature functional is (1) it allows us to interpret the singular/non-eulidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, and thus (2) one can compute the total curvature for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the curvature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embeddings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms.

A "clique minor" in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The "Hadwiger number" h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G*H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindrical grid with a vortex of bounded width in each of the two `big' faces, or - a toroidal grid. Motivation for studying the Hadwiger number of a graph includes Hadwiger's Conjecture, which states that the chromatic number chi(G)

The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. We also show that the logarithmic forest distances which are known to generalize the resistance distance and the shortest path distance are a subclass of walk distances. On the other hand, the long walk distance is equal to the resistance distance in a transformed graph.

In this paper we give an asymptotic expansion including error terms for the number of cycles in homology classes for connected graphs. Mainly, we obtain formulae about the coefficients of error terms which depend on the homology classes and give two examples of how to calculate the coefficient of first error term.

Introducing and studying the pattern frequency algebra, we prove the analogue of L\"uck's approximation theorems on $L^2$-spectral invariants in the case of aperiodic order. These results imply a uniform convergence theorem for the integrated density of states as well as the positivity of the logarithmic determinant of certain discrete Schrodinger operators.

Introducing and studying the pattern frequency algebra, we prove the analogue of L\"uck's approximation theorems on $L^2$-spectral invariants in the case of aperiodic order. These results imply a uniform convergence theorem for the integrated density of states as well as the positivity of the logarithmic determinant of certain discrete Schrodinger operators.

Introducing and studying the pattern frequency algebra, we prove the analogue of L\"uck's approximation theorems on $L^2$-spectral invariants in the case of aperiodic order. These results imply a uniform convergence theorem for the integrated density of states as well as the positivity of the logarithmic determinant of certain discrete Schrodinger operators.

For a hypergraph G and a positive integer s, let \chi_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter was studied by Kratochv\'{i}l, Tuza and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of \chi_{\ell} (G,s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs.

Rate two graphs for each of three scientific results.

If $K$ is a field with involution and $E$ an arbitrary graph, the involution from $K$ naturally induces an involution of the Leavitt path algebra $L_K(E).$ We show that the involution on $L_K(E)$ is proper if the involution on $K$ is positive definite, even in the case when the graph $E$ is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra $L_K(E)$ is regular if and only if $E$ is acyclic. We give necessary and sufficient conditions for $L_{K}(E)$ to be $^\ast$-regular (i.e. regular with proper involution). This characterization of $^\ast$-regularity of a Leavitt path algebra is given in terms of an algebraic property of $K,$ not just a graph-theoretic property of $E.$ This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph-theoretic properties of $E$ alone. As a corollary, we show that Handelman's conjecture (stating that every $^\ast$-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.

We consider the Anderson localization problem on the infinite regular tree. Within the localized phase, we derive a rigorous lower bound on the free energy function recently introduced by Aizenman and Warzel. Using a finite volume regularization, we also derive an upper bound on this free energy function. This yields upper and lower bounds on the critical disorder such that all states at a given energy become localized. These bounds are particularly useful in the large connectivity limit where they match, confirming the early predictions of Abou-Chacra, Anderson and Thouless.

For an integer $n\geq 2$, the triangular graph has vertex set the $2$-subsets of $\{1,\ldots,n\}$ and edge set the pairs of $2$-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every $2$-path lies in a unique quadrangle. We refine this result and provide a characterisation of connected locally triangular graphs as halved graphs of normal quotients of $n$-cubes. To do so, we study a parameter that generalises the concept of minimum distance for a binary linear code to arbitrary automorphism groups of the $n$-cube.

The cop number of a graph $G$ is the smallest $k$ such that $k$ cops win the game of cops and robber on $G$. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most $10g+15$ in case of orientable surface of genus $g$, and at most $10g'+15$ in case of non-orientable surface of Euler genus $g'$. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.

Delay-tolerant networks (DTNs) are characterized by a possible absence of end-to-end communication routes at any instant. Still, connectivity can generally be established over time and space. The optimality of a temporal path (journey) in this context can be defined in several terms, including topological (e.g. {\em shortest} in hops) and temporal (e.g. {\em fastest, foremost}). The combinatorial problem of computing shortest, foremost, and fastest journeys {\em given full knowledge} of the network schedule was addressed a decade ago (Bui-Xuan {\it et al.}, 2003). A recent line of research has focused on the distributed version of this problem, where foremost, shortest or fastest {\em broadcast} are performed without knowing the schedule beforehand. In this paper we show how to build {\em fastest} broadcast trees (i.e., trees that minimize the global duration of the broadcast, however late the departure is) in Time-Varying Graphs where intermittent edges are available periodically (it is known that the problem is infeasible in the general case even if various parameters of the graph are know). We address the general case where contacts between nodes can have arbitrary durations and thus fastest routes may consist of a mixture of {\em continuous} and {\em discontinuous} segments (a more complex scenario than when contacts are {\em punctual} and thus routes are only discontinuous). Using the abstraction of \tclocks to compute the temporal distances, we solve the fastest broadcast problem by first learning, at the emitter, what is its time of {\em minimum temporal eccentricity} (i.e. the fastest time to reach all the other nodes), and second by building a {\em foremost} broadcast tree relative to this particular emission date.

What happens with a graph if we multiply an argument of a function? In other words, how will a graph of a function y = f(k*x) look? Using similar approach, if a point (A,B) belongs to an original graph (i.e. B = f(A)) then (A/k,B) belongs to a graph of a new function. Indeed, let's substitute x = A/k and y = B to a new function: f(k*(A/k)) = f(A) = B which proves that (A/k,B) belongs to a graph of a new function. Geometrically, the fact that for each point (A,B) of an original graph there is a point (A/k,B) of a new graph means that an entire original graph is "squeezed" towards an Y-axis by a ratio of k. The word "squeeze" is really applicable to cases of k >= 1. When 0 < k < 1 it is really a "stretch". For k = -1 this transformation is a reflection relative to a Y-axis because for any point (A,B) on the original graph the point (-A,B) belongs to a graph of a new function y = f(-x). Finally, a transformation for any negative k is a reflection relative to a Y-axis and a horizontal "squeeze" or "stretch". How will a graph of a function change if we add a positive constant to a function? In other words, if you know how a graph of a function y = f(x) looks, what will be a graph of a function y = f(x) + a? Using a similar approach as before, assume that a point (A,B) belongs to an original graph (i.e. B = f(A)). Then, obviously, a point (A,B+a) belongs to a graph of a new function because it satisfies its algebraic expression: B+a = f(A) + a. It means, that a graph of a new function y = f(x) + a can be obtained by shifting an old graph up (for positive constant a) or down (for negative a). Finally, if we multiply a function by a constant (i.e. consider a new function y = k*f(x)) its graph will be "stretched" or "squeezed" in a vertical direction. The reason is, if a point (A,B) belongs to an original graph (i.e. B = f(A)) then (A,k*B) belongs to a new graph since k*f(A) = k*B

The classical zero-one law for first-order logic on random graphs says that for any first-order sentence $\phi$ in the theory of graphs, as n approaches infinity, the probability that the random graph G(n, p) satisfies $\phi$ approaches either 0 or 1. It is well known that this law fails to hold for any formalism that can express the parity quantifier: for certain properties, the probability that G(n, p) satisfies the property need not converge, and for others the limit may be strictly between 0 and 1. In this paper, we capture the limiting behavior of properties definable in first order logic augmented with the parity quantier, FO[parity], over G(n, p), thus eluding the above hurdles. Specifically, we establish the following "modular convergence law": For every FO[parity] sentence $\phi$, there are two rational numbers a_0, a_1, such that for i in {0,1}, as n approaches infinity, the probability that the random graph G(2n+i, p) satisfies $\phi$ approaches a_i. Our results also extend appropriately to first order logic equipped with Mod-q quantiers for prime q. Our approach is based on multivariate polynomials over finite fields, in particular, on a new generalization of the Gowers norm. The proof generalizes the original quantifier elimination approach to the zero-one law, and has analogies with the Razborov-Smolensky method for lower bounds for AC0 with parity gates.

We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk.

Graphs of Functions • Language: Hebrew • Author: Ashur Cherry (Toronto, Canada) Hebrew Language Publications are Dedicated in Honour of Professor Binyamin Geva

Graphs of Functions • Language: Hebrew • Author: Ashur Cherry (Toronto, Canada) Hebrew Language Publications are Dedicated in Honour of Professor Binyamin Geva

The probabilistic graphs framework models the uncertainty inherent in real-world domains by means of probabilistic edges whose value quantifies the likelihood of the edge existence or the strength of the link it represents. The goal of this paper is to provide a learning method to compute the most likely relationship between two nodes in a framework based on probabilistic graphs. In particular, given a probabilistic graph we adopted the language-constraint reachability method to compute the probability of possible interconnections that may exists between two nodes. Each of these connections may be viewed as feature, or a factor, between the two nodes and the corresponding probability as its weight. Each observed link is considered as a positive instance for its corresponding link label. Given the training set of observed links a L2-regularized Logistic Regression has been adopted to learn a model able to predict unobserved link labels. The experiments on a real world collaborative filtering problem proved that the proposed approach achieves better results than that obtained adopting classical methods.

We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite graphs. This bound is a generalization of the upper bound on the number of perfect matchings in bipartite graphs on $n+n$ vertices given by the Bregman-Minc inequality for the permanents of $(0,1)$ matrices.

A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide, to the best of our knowledge, the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families. This characterization implies an efficient recognition algorithm.

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let lambda be any nonnegative integer and mu be any positive integer. Suppose each pair of friends have exactly lambda common friends and each pair of strangers have exactly mu common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.