External Vertices Research Articles

Eigenvalue multiplicity of a graph in terms of the number of external vertices

The multiplicity of an eigenvalue λ of a graph G is denoted by m(G,λ). In a connected graph G with at least two vertices, a vertex is called external if it is not a cut vertex. In a tree, an external vertex is exactly a pendant vertex. Let ϵ(G) be the number of external vertices in G. In this paper, we prove that m(G,λ)≤ϵ(G)−1 for any λ and characterize the extremal graphs with m(G,−1)=ϵ(G)−1, which generalizes the main result of Wang et al. [Linear Multi. Alg., 2020] from a tree to an arbitrary connected graph.

External Vertices for Crystals of Affine Type A

External Vertices for Crystals of Affine Type A

Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks

The existence, uniqueness, strong and exponential stability of a generalized telegraph equation set on one dimensional star shaped networks are established. It is assumed that a dissipative boundary condition is applied at all the external vertices and an improved Kirchhoff law at the common internal vertex is considered. First, using a general criteria of Arendt-Batty (see Arendt and Batty in Trans. Am. Math. Soc. 306(2):837–852, 1988), combined with a new uniqueness result, we prove that our system is strongly stable. Next, using a frequency domain approach, combined with a multiplier technique and the construction of a new multiplier satisfying some ordinary differential inequalities, we show that the energy of the system decays exponentially to zero.

Two-Loop Kite Master Integral for a Correlator of Two Composite Vertices

We consider a two-loop propagator-type Feynman integral with two composite external vertices as a function of two Bjorken variables $$x$$ , $$y$$ , arbitrary indices $${{n}_{i}}$$ of propagators and space-time dimension $$D$$ . The integral is evaluated in terms of a hypergeometric double series. We find a chain of reductions of this double series to simpler functions in some physically interesting situations. The Mellin moments of the integral are also considered.

THE CATHEDRAL OF S. LORENZO IN PERUGIA AND THE HYPOGEAL SPACES. GEOMATIC TECHNIQUES FOR SPATIAL INVESTIGATIONS AIMED AT THE KNOWLEDGE AND INTERPRETATION OF THE ORIGIN OF THE TRANSEPT

Abstract. The area of the monumental complex of the Cathedral and the rectories of S. Lorenzo, located on the “acropolis” of Perugia, has been for more than two thousand years the main religious and civil reference site of all populations since the origin of the city of Perugia. The aim of this research was to survey the monumental complex of the Cathedral of San Lorenzo through the use of various geomatic techniques, with particular attention to the areas of the apse, the transept, the attic, the upper courtyard and the lower one, together with the hypogeal rooms of the Capitular Museum underlying the church, including important and impressive ruins as a portion of an Etruscan terracing wall (2nd century BC).The complex is extremely articulated, so it was necessary to correlate external and hypogeal internal spaces; the site is also characterized by the existence of numerous archaeological traces of various origins, resulting from the succession of events developed over the centuries and countless works that have been integrated, juxtaposed and in some cases replaced by previous ones. All this circumstances contribute to make not easy the understanding of the planimetric and altimetric relationships existing between the different parts and the identification of the real context of the architectural elements is equally complex. The need to connect different areas of the monumental complex located at different levels (from hypogeal rooms and vaults to the church, the attic and the external squares and streets) has required a coordinated and integrated use of geomatic techniques such as precision GNSS positioning (outside the building) and the creation of a very articulated three-dimensional geodetic network connecting the external GNSS vertices with the internal reference points and targets placed in the different areas to frame in a unique global datum the subsequent detailed surveys performed with LIDAR and photogrammetric techniques, so that the single scans and local surveys could be assembled to form a unique 3D model. Among the many aspects highlighted, in particular, it was possible to understand the genesis of the cathedral transept, whose size was dictated by an imposing Etruscan wall. Until now the ruins of the ancient cathedral complex were known - referring to three different buildings, the cathedral, the dodecagonal bell tower and the chapel of Sant'Ercolano - incorporated into the side of the basilical body in front of Piazza IV Novembre. From additional ruins attributed to the structure of the ancient cathedral, it was obtained that the level of the floor of the church has been substantially maintained in the current cathedral.

A Community Merger of Optimization Algorithm to Extract Overlapping Communities in Networks

A community in networks is a subset of vertices primarily connecting internal components, yet less connecting to the external vertices. The existing algorithms aim to extract communities of the topological features in networks. However, the edges of practical complex networks involving a weight that represents the tightness degree of connection and robustness, which leads a significant influence on the accuracy of community detection. In our study, we propose an overlapping community detection method based on the seed expansion strategy applying to both the unweighted and the weighted networks, called OCSE. First, it redefines the edge weight and the vertex weight depending on the influence of the network topology and the original edge weight, and then selects the seed vertices and updates the edges weight. Comparisons between OCSE approach and existing community detection methods on synthetic and real-world networks, the results of the experiment show that our proposed approach has the significantly better performance in terms of the accuracy.

A topological framework for signed permutations

In this paper, we present a topological framework for studying signed permutations and their reversal distance. This framework is based on a presentation of orientable and non-orientable fatgraphs via sectors. As an application, we give an alternative approach and interpretation of the Hannenhalli–Pevzner formula for the reversal distance of sorting signed permutations. This is obtained by constructing a bijection between signed permutations and certain equivalence classes of fatgraphs, called π-maps. We study the action of reversals and show that they either splice, glue or half-flip external vertices, which implies that any reversal changes the topological genus by at most one. We show that the lower bound of the reversal distance of a signed permutation equals the topological genus of its π-maps. We then discuss how the new topological model connects to other sorting problems.

Control and stabilization of 2 × 2 hyperbolic systems on graphs

We consider 2× 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.

Stability of non-autonomous difference equations with applications to transport and wave propagation on networks

In this paper, we address the stability of transport systems and wave propagation on networks with time-varying parameters. We do so by reformulating these systems as non-autonomous difference equations and by providing a suitable representation of their solutions in terms of their initial conditions and some time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of such coefficients. In the case of difference equations with arbitrary switching, we obtain a delay-independent generalization of the well-known criterion for autonomous systems due to Hale and Silkowski. As a consequence, we show that exponential stability of transport systems and wave propagation on networks is robust with respect to variations of the lengths of the edges of the network preserving their rational dependence structure. This leads to our main result: the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but at most one.

Search algorithm of structure anomalies in complete graph based on scattering quantum walk

Quantum walks have been proven to be a useful framework in designing new quantum algorithms, of which the search algorithm is the most notable. Besides a general search for a special vertex, recent researches have shown that quantum walks can also be used to find structural anomalies. Suppose a vertex of complete graph KN is attached to a second graph G, then the kind of structure anomaly will break the symmetry of the complete graph. The search algorithm based on scattering quantum walk model is presented to speed up locating this kind of structure anomaly. The concepts of scattering quantum walk model and collapsed graphs are presented. The definition of the evolutionary operator, which is different from that of a general search, is given. Based on the specific definition of evolutionary operator and the obvious symmetry of complete graph, it is shown that the search space is confined to a low-dimensional collapsed space, and the initial state is chosen to lie in this subspace. To illustrate the evolutionary process of the search algorithm, an example is given in the case that G is a single vertex. Taking advantage of our earlier study on the evolutionary operator of coined quantum walks with Grover coin, calculations of the unitary operator in the collapsed space are greatly simplified. To quantify the evolutionary process of the algorithm, we use the matrix perturbation theory involving a perturbative approach to find the eigenvalues and eigenstates. It is the degenerate zeroth-order eigenvalue 0 = 1 that leads to the interesting parts of the Hilbert space. Most of the recent researches of searching the structure anomalies focus on star graph SN with an unspecified graph G attached to one of its external vertices, where the overall graph is divided into two parts by the central vertex. It is shown that quantum speedup will occur if and only if the eigenvalues associated with these two parts in the N limit are the same. In this paper, we find that the collapsed graph of complete graphs can also be divided into two parts by a single collapsed vertex. As these two parts roughly correspond to the initial state and the desired state respectively, the techniques and results in star graphs can be generalized to the collapse graph on complete graph. What is more, under our definition of unitary evolution operator these two parts in the N limit will always share the same eigenvalue, i.e. 0 = 1, no matter what the structure of graph G is. Based on this, we prove that the search algorithm can find the target vertex in O(N) time steps with a success probability of 1-O(1N). That is to say, the quantum search algorithm gains a quadratic speedup over classical counterpart.

Inverse eigenvalue problem for a simple star graph

A Schrödinger operator and associated spectra may be defined for a graph by identifying edges with intervals of \mathbb R , on which coefficient functions are defined, imposing appropriate matching conditions at the internal vertices and boundary conditions at the external vertices. Following earlier work of Pivovarchik [14], we consider an inverse eigenvalue problem for a graph consisting of three equal length edges meeting at a single point, where the spectral data is the Dirichlet eigenvalues of the graph together with the Dirichlet spectra of the three individual edges. We derive, discuss and demonstrate a constructive solution method, obtain an alternative uniqueness proof, and discuss several kinds of generalizations.

Evaluating single-scale and/or non-planar diagrams by differential equations

We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, $p_2^2\neq 0$. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with $\epsilon=(4-D)/2$. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small $p_2^2$ to our results at $p_2^2\neq 0$ and to identify various terms in these solutions according to expansion by regions. The second family consists of four-point non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called $K_4$ graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in $\epsilon$ up to weight six.

Quantum walks as a probe of structural anomalies in graphs

We study how quantum walks can be used to find structural anomalies in graphs via several examples. Two of our examples are based on star graphs, graphs with a single central vertex to which the other vertices, which we call external vertices, are connected by edges. In the basic star graph, these are the only edges. If we now connect a subset of the external vertices to form a complete subgraph, a quantum walk can be used to find these vertices with a quantum speedup. Thus, under some circumstances, a quantum walk can be used to locate where the connectivity of a network changes. We also look at the case of two stars connected at one of their external vertices. A quantum walk can find the vertex shared by both graphs, again with a quantum speedup. This provides an example of using a quantum walk in order to find where two networks are connected. Finally, we use a quantum walk on a complete bipartite graph to find an extra edge that destroys the bipartite nature of the graph.

Global Carleman estimate on a network for the wave equation and application to an inverse problem

We are interested in an inverse problem for the wave equation with potential on a starshaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.

Some inverse scattering problems on star-shaped graphs

Having in mind applications to the fault-detection/diagnosis of lossless electrical networks, here we consider some inverse scattering problems for Schrödinger operators over star-shaped graphs. We restrict ourselves to the case of minimal experimental setup consisting in measuring, at most, two reflection coefficients when an infinite homogeneous (potential-less) branch is added to the central node. First, by studying the asymptotic behavior of only one reflection coefficient in the high-frequency limit, we prove the identifiability of the geometry of this star-shaped graph: the number of edges and their lengths. Next, we study the potential identification problem by inverse scattering, noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings (potentials with bounded H 1 -norms). The main result states that, under some assumptions on the geometry of the graph, the measurement of two reflection coefficients, associated to two different sets of boundary conditions at the external vertices of the tree, determines uniquely the potentials; it can be seen as a generalization of the theorem of the two boundary spectra on an interval.

The Aldous-Shields model revisited with application to cellular ageing

In Aldous and Shields (1988) a model for a rooted, growing random binary tree with edge lengths 1 was presented. For some $c>0$, an external vertex splits at rate $c^{-i}$ (and becomes internal) if its distance from the root (depth) is $i$. We reanalyse the tree profile for $c>1$, i.e. the numbers of external vertices in depth $i=1,2,...$. Our main result are concrete formulas for the expectation and covariance-structure of the profile. In addition, we present the application of the model to cellular ageing. Here, we say that nodes in depth $h+1$ are senescent, i.e. do not split. We obtain a limit result for the proportion of non-senesced vertices for large $h$.

Retarded Propagator Representation of Out-of-Equilibrium Thermal Field Theories

We represent out of equilibrium thermal field theories with finite time path in terms of retarded propagators exclusively. For the particle number, defined as the equal time limit of the Keldysh propagator, the time ordering of the diagrams contributing is particularly simple: all external end-points of propagators have maximal time, there are no internal vertices with locally maximal time, the property which guaranties causality), there is, at least one “sink” vertex (vertex with locally minimal time). The diagram looks like fisher net hanging on external vertices. At the “sink” vertices energy is not conserved, thus establishing realisation of uncertainty relations in out of equilibrium TFT. Even more, at the equal-time limit, the terms conserving energy at “sink” vertices vanish. This fact eliminates pinching problem and enables safe t i m e → ∞ limit. The retarded propagator in higher orders is regularized only as a part of of the diagram connected to equal time limit of multi point Green function representing expectation value of the product of number operators. These properties indicate clear advantage of finite time path, in large time limit over the use of Keldysh time path.

Role of the UL25 Protein in Herpes Simplex Virus DNA Encapsidation

The herpes simplex virus protein UL25 attaches to the external vertices of herpes simplex virus type 1 capsids and is required for the stable packaging of viral DNA. To define regions of the protein important for viral replication and capsid attachment, the 580-amino-acid UL25 open reading frame was disrupted by transposon mutagenesis. The UL25 mutants were assayed for complementation of a UL25 deletion virus, and in vitro-synthesized protein was tested for binding to UL25-deficient capsids. Of the 11 mutants analyzed, 4 did not complement growth of the UL25 deletion mutant, and analysis of these and additional mutants in the capsid-binding assay demonstrated that UL25 amino acids 1 to 50 were sufficient for capsid binding. Several UL25 mutations were transferred into recombinant viruses to analyze the effect of the mutations on UL25 capsid binding and on DNA cleavage and packaging. Studies of these mutants demonstrated that amino acids 1 to 50 of UL25 are essential for its stable interaction with capsids and that the C terminus is essential for DNA packaging and the production of infectious virus through its interactions with other viral packaging or tegument proteins. Analysis of viral DNA cleavage demonstrated that in the absence of a functional UL25 protein, aberrant cleavage takes place at the unique short end of the viral genome, resulting in truncated viral genomes that are not retained in capsids. Based on these observations, we propose a model where UL25 is required for the formation of DNA-containing capsids by acting to stabilize capsids that contain full-length viral genomes.

Single Crystal X-ray Diffraction Study of a Pure Natrolite Sample

The crystal structure of a pure natrolite sample, Na2(Al2Si3O10)·2H2O, coming from Asheken, Ethiopia, has been analysed by single crystal X-ray diffraction. It crystallizes within the orthorhombic space group Fdd2, with the following cell constants: a = 18.2930(2) A; b = 18.6430(5) A; c = 6.5860(5) A; V = 2246.07(18) A3. The three-dimensional framework of this hydrated aluminosilicate zeolite is made up by chains of corner-sharing SiO4 and AlO4 tetrahedra down c; the chains are held together by sharing the external vertices of tetrahedra; water molecules and Na+ extra-framework cations fill up the resulting cavities, the latter forming irregular NaO6 octahedra. Hydrogen bonds complete the array.

Calculation of Feynman diagrams in space and time by stationary phase

AbstractFeynman diagrams are usually derived in relativistic space and time, and then immediately converted into energy and momentum by a Fourier transform. A method is proposed that allows keeping the external vertices at fixed locations and finite times, while the internal vertices are integrated in space‐time as well. Propagators between the vertices are the well‐known Bessel functions of the relativistic distance. The example of the Compton effect is worked out.