Arbitrary Vertex Research Articles

Hardness of Approximate Diameter: Now for Undirected Graphs

Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open whether a better approximation is possible in near-linear time. A series of papers on fine-grained complexity have led to strong hardness results for diameter in directed graphs, culminating in a recent tradeoff curve independently discovered by [Li, STOC’21] and [Dalirrooyfard and Wein, STOC’21], showing that under the Strong Exponential Time Hypothesis (SETH), for any integer k ≥ 2 and δ > 0, a \(2-\frac{1}{k}-\delta \) approximation for diameter in directed m -edge graphs requires m 1 + 1/( k − 1) − o (1) time. In particular, the simple linear time 2-approximation algorithm is optimal for directed graphs. In this paper we prove that the same tradeoff lower bound curve is possible for undirected graphs as well, extending results of [Roditty and Vassilevska W., STOC’13], [Li’20] and [Bonnet, ICALP’21] who proved the first few cases of the curve, k = 2, 3 and 4, respectively. Our result shows in particular that the simple linear time 2-approximation algorithm is also optimal for undirected graphs. To obtain our result, we extract the core ideas in known reductions and introduce a unification and generalization that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation.

Bridge Sampling for Connections via Multiple Scattering Events

AbstractExplicit sampling of and connecting to light sources is often essential for reducing variance in Monte Carlo rendering. In dense, forward‐scattering participating media, its benefit declines, as significant transport happens over longer multiple‐scattering paths around the straight connection to the light. Sampling these paths is challenging, as their contribution is shaped by the product of reciprocal squared distance terms and the phase functions. Previous work demonstrates that sampling several of these terms jointly is crucial. However, these methods are tied to low‐order scattering or struggle with highly‐peaked phase functions.We present a method for sampling a bridge: a subpath of arbitrary vertex count connecting two vertices. Its probability density is proportional to all phase functions at inner vertices and reciprocal squared distance terms. To achieve this, we importance sample the phase functions first, and subsequently all distances at once. For the latter, we sample an independent, preliminary distance for each edge of the bridge, and afterwards scale the bridge such that it matches the connection distance. The scale factor can be marginalized out analytically to obtain the probability density of the bridge. This approach leads to a simple algorithm and can construct bridges of any vertex count. For the case of one or two inserted vertices, we also show an alternative without scaling or marginalization.For practical path sampling, we present a method to sample the number of bridge vertices whose distribution depends on the connection distance, the phase function, and the collision coefficient. While our importance sampling treats media as homogeneous we demonstrate its effectiveness on heterogeneous media.

Towards systematic evaluation of de Sitter correlators via Generalized Integration-By-Parts relations

We generalize Integration-By-Parts (IBP) and differential equations methods to de Sitter correlators related to inflation. While massive correlators in de Sitter spacetime are usually regarded as highly intricate, we find they have remarkably hidden concise structures from the perspective of IBP. We find the factorization of the IBP relations of each vertex integral family corresponding to dτi integration. Furthermore, with a smart construction of master integrals, the universal formulas for iterative reduction and d log-form differential equations of arbitrary vertex integral family are presented and proved. These formulas dominate all tree-level de Sitter correlators and play a kernel role at the loop-level as well.

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Exotic eigenvalues of shrinking metric graphs

Eigenvalue spectrum of the Laplacian on a metric graph with arbitrary but fixed vertex conditions is investigated in the limit as the lengths of all edges decrease to zero at the same rate. It is proved that there are exactly four possible types of eigenvalue asymptotics. The number of eigenvalues of each type is expressed via the index and nullity of a form defined in terms of the vertex conditions.

Elusive properties of infinite graphs

AbstractA graph property is said to be elusive (or evasive) if every algorithm testing this property by asking questions of the form “is there an edge between vertices and ?” requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa–Karp–Rosenberg conjecture is that every nontrivial monotone graph property is elusive for any finite vertex set. We show that the situation is completely different for infinite vertex sets: the monotone graph properties “every vertex has degree at least ” and “every connected component has size at least ,” where and are natural numbers, are not elusive for infinite vertex sets, but the monotone graph property “the graph contains a cycle” is elusive for arbitrary vertex set. On the other hand, we also prove that every algorithm testing some natural monotone graph properties, for example, “every vertex has degree at least ” or “connected” on the vertex set should check “lots of edges,” more precisely, all the edges of an infinite complete subgraph.

Quantum walk state transfer on a hypercube

We investigate state transfer on a hypercube by means of a quantum walk where the sender and the receiver vertices are marked by a weighted loops. First, we analyze search for a single marked vertex, which can be used for state transfer between arbitrary vertices by switching the weighted loop from the sender to the receiver after one run-time. Next, state transfer between antipodal vertices is considered. We show that one can tune the weight of the loop to achieve state transfer with high fidelity in shorter run-time in comparison to the state transfer with a switch. Finally, we investigate state transfer between vertices of arbitrary distance. It is shown that when the distance between the sender and the receiver is at least 2, the results derived for the antipodes are well applicable. If the sender and the receiver are direct neighbours the evolution follows a slightly different course. Nevertheless, state transfer with high fidelity is achieved in the same run-time.

One-loop contributions to decays eb → eaγ and [formula omitted] anomalies, and Ward identity

In this paper, we will present analytic formulas to express one-loop contributions to lepton flavor violating decays eb→eaγ, which are also relevant to the anomalous dipole magnetic moments of charged leptons ea. These formulas were computed in the unitary gauge, using the well-known Passarino-Veltman notations. We also show that our results are consistent with those calculated previously in the 't Hooft-Veltman gauge, or in the limit of zero lepton masses. At the one-loop level, we show that the appearance of fermion-scalar-vector type diagrams in the unitary gauge will violate the Ward Identity relating to an external photon. As a result, the validation of the Ward Identity guarantees that the photon always couples with two identical particles in an arbitrary triple coupling vertex containing a photon.

Light subpath reservoir for interactive ray-traced global illumination

In this paper, we propose a novel approach for sampling indirect illumination based on bidirectional path tracing and world space resampling reservoir. We cache light subpaths with high contribution in a world space grid, allowing for sampling at arbitrary vertices along the eye path. In order to find optimal samples, we continuously update the reservoirs stored in each cell according to the resampling principles. Experiments have shown that our algorithm achieves noticeable noise reduction in scenes lit by indirect light compared with path tracing, at the cost of additional time and memory consumption, while still maintaining interactive frame rates. In some special cases, our method even shows better results than a plain bidirectional path tracer, in terms of both image quality and performance.

Two Channel Filter Banks on Arbitrary Graphs With Positive Semi Definite Variation Operators

We propose novel two-channel filter banks for signals on graphs. Our designs can be applied to arbitrary graphs, given a positive semi definite variation operator, while using arbitrary vertex partitions for downsampling. The proposed generalized filter banks (GFBs) also satisfy several desirable properties including perfect reconstruction and critical sampling, while having efficient implementations. Our results generalize previous approaches that were only valid for the normalized Laplacian of bipartite graphs. Our approach is based on novel graph Fourier transforms (GFTs) given by the generalized eigenvectors of the variation operator. These GFTs are orthogonal in an alternative inner product space which depends on the downsampling and variation operators. Our key theoretical contribution is showing that the spectral folding property of the normalized Laplacian of bipartite graphs, at the core of bipartite filter bank theory, can be generalized for the proposed GFT if the inner product matrix is chosen properly. In addition, we study vertex domain and spectral domain properties of GFBs and illustrate their probabilistic interpretation using Gaussian graphical models. While GFBs can be defined given any choice of a vertex partition for downsampling, we propose an algorithm to optimize these partitions with a criterion that favors balanced partitions with large graph cuts, which are shown to lead to efficient and stable GFB implementations. Our numerical experiments show that partition-optimized GFBs can be implemented efficiently on 3D point clouds with hundreds of thousands of points (nodes), while also improving the color signal representation quality over competing state-of-the-art approaches.

Reliability assessment of the divide-and-swap cube in terms of generalized connectivity

The generalized connectivity of a graph G, denoted as κk(G), is a generalization of the traditional connectivity and is a parameter to measure the capability of connecting multiple vertices in G. The divide-and-swap cube DSCn is a variant of the hypercube with a nice hierarchical structure and plentiful properties. This paper investigates the generalized connectivity on DSCn. We obtain the result κ4(DSCn)=d, where d=log2⁡n≥1, by the construction showing that there are d internally disjoint trees connecting any four arbitrary vertices of DSCn. As a corollary, one can directly obtain κ3(DSCn)=d.

Semihypergroup-Based Graph for Modeling International Spread of COVID-n in Social Systems

Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic application in some problems. Influenced by these mathematical notions, a novel semihypergroup-based graph (SBG) of G=H,E is constructed through the fundamental relation γn on H, where semihypergroup H is appointed as the set of vertices and E is addressed as the set of edges on SBG. Indeed, two arbitrary vertices x and y are adjacent if xγny. The connectivity of graph G is characterized by xγ*y, whereby the connected components SBG of G would be exactly the elements of the fundamental group H/γ*. Based on SBG, some fundamental characteristics of the graph such as complete, regular, Eulerian, isomorphism, and Cartesian products are discussed along with illustrative examples to clarify the relevance between semihypergroup H and its corresponding graph. Furthermore, the notions of geometric space, block, polygonal, and connected components are introduced in terms of the developed SBG. To formulate the links among individuals/countries in the wake of the COVID (coronavirus disease) pandemic, a theoretical SBG methodology is presented to analyze and simplify such social systems. Finally, the developed SBG is used to model the trend diffusion of the viral disease COVID-n in social systems (i.e., countries and individuals).

Structure of Projective Planar Subgraphs of the Graph Obstructions for Fixed Surface

Consider the problem of studying the metric properties of a subgraph G\v, where v is an arbitrary vertex of obstruction graphs G of a nonorientable genus, which will determine the sets of points of attachment of one subgraph to another and allow constructing prototypes of graphs-obstruction with number of vertices greater than 10 nonorientable genus greater than 1. This problem is related to Erdosh's hypothesis [3] on the coverage of obstruction graphs of an undirected surface of the genus k, where k > 0, the smallest inclusion of the set of k + 1st graph of the homeomorphic K5, or K3,3, in [4] constructively proved for 35 minors of obstruction graphs of the projective plane, a set of 62 with no more than 10 vertices of obstruction graphs and their splits for the Klein surface, as well as some obstruction graphs for other surfaces. In [5], the existence of a finite set of obstruction graphs for a non-orienting surface was proved. A similar problem was considered in [6], where models or prototypes of obstruction graphs were considered. The prototype of the graph-obstruction of the undirected genus, we will call the graphs that have their own subgraph graph-obstruction of the undirected genus. In [7, 8] the tangent problem of covering the set of vertices with the smallest number of cycles-boundaries of 2-cells was considered, the concept of cell distance is given in [9, 10], where the boundaries of an oriented genus of graphs formed from planar graphs and a simple star glued to some of its peaks. Hypothetically, it is possible to obtain them by recursive φ-transformation of the graph-obstruction of the projective plane and a copy of its planar subgraph given on vertices, edges or parts of edges, or simple chains, i.e. achievable parts of the so-called graph-basis (graph of homeomorphic graph Kuratovsky plane). We assume that instead of one subgraph there can be several copies of subgraphs of graphs-obstructions of the projective plane.The article has an introduction and two parts, in which the structural properties of subgraphs of obstruction graphs for an undirected surface, presented as a φ-image of one of the Kuratovsky graphs and at least one planar graph, are investigated. The metric properties of the minimal embeddings of the subgraphs of the obstruction graphs for undirected surfaces are considered, and the main result is Theorems 1, 2, and Lemma 3 as the basis of the prototype construction algorithm. Keywords: φ-transformation of graphs, nonorientable surface, prototypes of graph-obstruction.

AN ANALYTIC INVERSION FORMULA FOR A RADON TRANSFORM ON A CLASS OF CONES

Since the works of Radon and Cormack on the classical Radon transform on straight lines, several generalizations involving integrations on conical surfaces have been studied. In this article, we introduce a new family of conical surfaces with arbitrary vertices of the space and axes through the origin and study the corresponding Radon transform. We derive its analytical inversion in two and three dimensions. Numerical simulations are carried out for the two-dimensional case.

Shear horizontal wave propagation in inhomogeneous wedge space

In this paper, a closed-form analytical solution is presented for SH wave propagation in density radial inhomogeneous wedge space. The material parameter of this inhomogeneous wedge space with arbitrary vertex angle is given in functional form. Based on complex function method, an appropriate mapping function is introduced to transform the governing Helmholtz equation with variable coefficients into a standard one. The wave field expression satisfying zero-stress boundary condition in wedge space is derived. Finally, numerical examples are presented to analysis the influence of different parameters on displacement amplitudes.

Generating Integrally Indecomposable Newton Polygons with Arbitrary Many Vertices

In this paper we shall give another proof of a special case of Gao’s theorem for generating integrally indecomposable polygons in the sense of Minkowski. The approach of proving this theorem will enable us to give an effective algorithm for construction integrally indecomposable convex integral polygons with arbitrary many vertices. In such a way, classes of absolute irreducible bivariate polynomials corresponding to those indecomposable Newton polygons are generated.

Monophonic pebbling number and t-pebbling number of some graphs

Assume G is a graph with some pebbles distributed over its vertices. A pebbling move is when two pebbles are removed from one vertex, one is thrown away, and the other is moved to an adjacent vertex. The monophonic pebbling number, of a connected graph G, is the least positive integer n such that any distribution of n pebbles on G allows one pebble to be carried to any specified but arbitrary vertex using monophonic path by a sequence of pebbling operations. The least positive integer n such that any distribution of n pebbles on G allows t pebbles to be moved to any specified but arbitrary vertex by a sequence of pebbling moves using monophonic path is the monophonic t-pebbling number The monophonic pebbling number and monophonic t-pebbling number of Jahangir graphs, paths and square of paths are determined in this study.

Sum divisor cordial labeling in the context of graph operations on grötzsch

A Sum divisor cordial labeling of a graph G with vertex set V is a bijection r from V to {1,2,3,...,|V (G )|} such that an edge uv is assigned the label 1 if 2 divides r(u)+ r (v ) and 0 otherwise; and the number of edges labeled with 0 and the number of edges labeled with 1differ by at most 1 . A graph with a sum divisor cordial labeling is called sum divisor cordial graph. In this research paper, we investigate the sum divisor cordial labeling bahevior for Grötzsch graph, fusion of any two vertices in Grötzsch graph, duplication of an arbitrary vertex in Grötzsch graph, duplication of an arbitrary vertex by an edge in Grötzsch graph, switching of an arbitrary vertex of degree four in Grötzsch graph, switching of an arbitrary vertex of degree three in Grötzsch graph and path union of two copies of Grötzsch.

Computation of edge Pi index, vertex Pi index and Szeged index of some cactus chains

A cactus chain is a connected graph in which all blocks are cycles, each cycle has at most two cut-vertices and each cut-vertex is shared by exactly two cycles. In this paper we give exact values of edge PI index and vertex PI index of an arbitrary cactus chain and vertex Szeged index of some special types of cactus chains.

Efficient optimization of the Held–Karp lower bound

Given a weighted undirected graph G=(V,E), the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex p ¯∈V, by computing a minimum cost tree spanning V∖{p ¯} and adding two minimum cost edges adjacent to p ¯. In general, different selections of vertex p ¯ provide different lower bounds. In this paper it is shown that the selection of vertex p ¯ can be optimized, to obtain the largest possible Held–Karp lower bound, with the same worst-case computational time complexity required to compute a single minimum spanning tree. Although motivated by the optimization of the Held–Karp lower bound for the TSP, the algorithm solves a more general problem, allowing for the efficient pre-computation of alternative minimum spanning trees in weighted graphs where any vertex can be deleted.