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The eccentricity matrix ɛ (G) of a graph G is obtained from the distance matrix D (G) by retaining only for each row and each column the largest distance, and setting the remaining elements as 0. In this paper, we firstly show that the eccentricity matrix of clique trees is irreducible. We identify the t -clique trees with given diameter odd d having the maximum ɛ -spectral radius, and the corresponding extremal graphs are also determined. We determine the upper bounds for the ɛ -spectral radius of t -clique trees which d and n satisfy that one is odd and the other is even. Finally, we propose some potential topics for further study. [ABSTRACT FROM AUTHOR]

The microsphere profile of probe tips has to be measured and compensated to improve the measurement accuracy of micro-coordinate measuring machines (micro-CMMs) to less than several hundred nanometers. A high-precision radius and sphericity measurement method for the microsphere of the CMM probe tip is proposed in this article. Different circumferences of the tested microsphere are measured, and the true radius of any measuring point on the surface can be obtained after separating the runout errors. Then the sphericity of the tested microsphere can be evaluated by the minimum zone sphere method. A corresponding measuring system is developed based on the analysis of the primary error model and mechanical model, and verification experiments are conducted using a ruby microsphere (A-5000-7801, Renishaw Corporation) as the reference, whose claimed diameter and sphericity are 700 µm and 130 nm, respectively. Six groups of repeated experiments are performed, and 18 000 measurement points on 15 circumferences are recorded in each group of experiments. Results show that the average radius of the tested microsphere is 350.003 µm, and the average sphericity error is 208 nm with a standard deviation of 5.3 nm. Finally, the expanded uncertainty of the measurement results of the developed system is calculated as 209 nm (k = 1.96, normal distribution). The proposed method and system can be used in the precision measurements and compensations for probe tips of micro-CMMs and other parts, such as microspheres used in micro-bearings and micro-manufacturing machines. [ABSTRACT FROM AUTHOR]

The local metric dimension is one of many topics in graph theory with several applications. One of its applications is a new model for assigning codes to customers in delivery services. Let G be a connected graph and V(G) be a vertex set of G. For an ordered set W = fx1; x2;: :: ; xkg - V(G), the representation of a vertex x with respect to W is rG(xjW) = f(d(x; x1); d(x; x2);: :: ; d(x; xk)g. The set W is said to be a local metric set of G if r(xjW), r(yjW) for every pair of adjacent vertices x and y in G. The eccentricity of a vertex x is the maximum distance between x and all other vertices in G. Among all vertices in G, the smallest eccentricity is called the radius of G and a vertex whose eccentricity equals the radius is called a central vertex of G. In this paper, we developed a new concept, so-called the central local metric dimension by combining the concept of local metric dimension with the central vertex of a graph. The set W is a central local metric set if W is a local metric set and contains all central vertices of G. The minimum cardinality of a central local metric set is called a central local metric dimension of G. In the main result, we introduce the definition of the central local metric dimension of a graph and some properties, then construct the central local metric dimensions for trees and establish results for the grid graph. [ABSTRACT FROM AUTHOR]

Copyright of Bearing is the property of Bearing Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of China University of Petroleum is the property of China University of Petroleum and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper presents the theoretical and experimental results of a study investigating a new method for tube bending. The method involves three-point bending of a tube without using a mandrel. The bending process was conducted with a small bending radius of Rg = 1.5Dz (where Dz is the outside diameter of the bent tube) and a large bending angle of 180°. The novelty of the proposed solution is the use of new shapes of bending roll impression. Instead of the standard circular-shaped impression, an elliptical-shaped impression was used. The aim of the study on the proposed small radius tube bending technique was to optimize the shape of roll impression in terms of minimizing ovalization and flattening of the tube cross section in the bending zone. Previous studies only showed that circular impressions were inefficient. The tube bending process conducted with a circular impression roll, without the use of a mandrel or other type of filling to achieve an angle ranging 90°÷180°, led to the flattening of the cross section. The tube wall in the upper zone would crack or its cross section would become deformed and oval. This theoretical and experimental study was conducted on tubes with an outside diameter of Dz = 20 mm and a wall thickness of g = 2 mm, made of 16Mo3 boiler steel and EN-AW 6060 aluminum alloy. Obtained results were then used to determine the ranges of bending roll impression parameters that ensured that the product would meet the standardized conditions of cross-sectional ovalization in the bending zone. The tool developed for this study can be applied in industrial practice. [ABSTRACT FROM AUTHOR]

For a connected graph G (V , E) , we use the notation d (a 1 , a 2) to represent the distance between two node points a 1 and a 2 and it is the minimum of the lengths of all paths between them. The eccentricity e (a) of a node point a ∈ V is considered as the maximum length of all shortest paths starts from a to the remaining nodes, i.e., e (a) = max { d (a , u 1) : u 1 ∈ V }. The diameter of a graph G , we denote it by d (G) and it is the length of the longest shortest path in G , i.e., d (G) = max { e (a) : a ∈ V }. Also, the radius of a graph G , we denote it by the symbol r (G) and it is the least eccentricity of all node points in V , i.e., r (G) = min { e (a) : a ∈ V }. The central vertex/node point x of a graph G is a node whose eccentricity is same as G 's radius, i.e., e (x) = r (G). The collection of all central nodes of a graph G is considered as the center of G and it is symbolized by c (G) , i.e., c (G) = { x ∈ V : e (x) = r (G) }. A graph may have one or more central vertices. This paper develops an optimal algorithm to compute the diameter, radius and central node (s) of the permutation graph having n node points in O (n) time. We have also established a tight relation between radius and diameter of permutation graphs. [ABSTRACT FROM AUTHOR]

The problem of calculating the moisture content of pore space of sediment after its mechanical drying in a centrifuge with a peripheral filtering surface is studied. An analytical equation was derived along the radius of moisture filling of the sediment by filtrate of the capillary fringe, taking account of lognormal distribution of pore sizes. The proportion of moisture imported into the pores by the bound filtrate is determined. The presence of a basal layer of the capillary fringe is illustrated. The possibility of limiting determination of the smallest pore diameter toward larger size for constructing histograms and calculating the moisture content of the sediment is confirmed. The agreement of the results with the known experimental data received for suspensions of chemical and oil refining industries in filtering centrifuges is confirmed by an example of calculation. The main variable design and technological parameters for calculations based on effective mechanical drying of sediment in the filtering centrifuge rotor are the radius of the circumference of its filtering surface, the angular velocity, and the maximum size of pores in the sediment. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 1, a k-distance enclaveless number (or k-distance B-differential) of a connected graph G = (V, E) is Ψ k (G) = max{|(V − X) ∩ Nk,G(X)| : X ⊆ V }. In this paper, we establish upper bounds on the k-distance enclaveless number of a graph in terms of its diameter, radius and girth. Also, we prove that for connected graphs G and H with orders n and m respectively, Ψk (G × H) ≤ mn − n − m + Ψk (G) + Ψk (H) + 1, where G × H denotes the direct product of G and H. In the end of this paper, we show that the k-distance enclaveless number Ψk (T) of a tree T on n ≥ k + 1 vertices and with n1 leaves satisfies inequality Ψ k (T) ≤ k(2n−2+n1) 2k+1 and we characterize the extremal trees. [ABSTRACT FROM AUTHOR]

In low-permeability reservoirs, the average pore throat radius is small and the heterogeneity is severe, and the threshold pressure gradient exists obviously. Because of the pore throat radius sorting of porous media, the threshold pressure gradients of the pore throats with different radius are different, which leads to the asynchronism of the fluid participating in flow in the pore throats with different radius. The apparent permeability of the rock is a variable value before it reaches the average permeability of the rock, which makes low-velocity nonlinear porous flow come into being. The essence of nonlinear porous flow is the process of the fluid participating in porous flow gradually according to pore throat radius from large to small in pore throats with different radius under the combined action of threshold pressure gradient and pore throat radius sorting. This is an inevitable stage before the porous media reaches Darcy flow. The porous media keeps in nonlinear porous flow stage before all the pore throats participate in flowing. Based on the pore throat radius sorting of porous media, the nonlinear porous flow equation in low-permeability reservoirs was established using unequal-diameter parallel capillary bundle model with the capillary radius commonly in accordance with normal distribution. The equation can not only reveal low-velocity nonlinear porous flow mechanism but also describe the forming process of nonlinear porous flow. Furthermore, the equation is a general equation. Both Darcy's law and the pseudo-Darcy equation with the threshold pressure gradient in low-permeability reservoirs are the particular cases. [ABSTRACT FROM AUTHOR]

Bearing skidding is likely to cause early failure of the bearings. Traditional methods of bearing skidding analysis claim that the bearing parameters are determined, but they are random in actual working conditions, which may cause larger analysis errors. In order to solve this problem- a reliability analysis model of angular contact ball bearing skidding which takes into account the influence of random factors is proposed. According to the quasi-static analysis of rolling bearings and the criterion of Hirano for bearing skidding, the critical surface of skidding is given, and the limit-state equation of bearing skidding is established based on whether the bearing is skidding or not. The reliability sensitivity analysis is carried out by using Kriging method to evaluate the influence of bearing parameters on bearing skidding. The research results show that the change of ball diameters has the most impact on the bearing skidding, then the raceway diameter and groove curvature radius of the inner and outer raceway take the second place, and the contact angle is the least. The research results may provide the theoretical basis for reducing or avoiding ball bearing skidding. [ABSTRACT FROM AUTHOR]

A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. Harary and Thomassen ['Anticritical graphs', Math. Proc. Cambridge Philos. Soc.79(1) (1976), 11–18] proved that the radius r and diameter d of any radially maximal graph satisfy $r\le d\le 2r-2.$ Dutton et al. ['Changing and unchanging of the radius of a graph', Linear Algebra Appl.217 (1995), 67–82] rediscovered this result with a different proof and conjectured that the converse is true, that is, if r and d are positive integers satisfying $r\le d\le 2r-2,$ then there exists a radially maximal graph with radius r and diameter $d.$ We prove this conjecture and a little more. [ABSTRACT FROM AUTHOR]

While many studies on planar elastomeric bearings have attracted attention in the international literature, there are very few studies on spherical elastomeric bearings due to their multi-layered and design difficulties. While elastomeric bearings are rigid against loads perpendicular to the layers, they are flexible against the loads parallel to the layers. In this way, spherical elastomeric bearings provide rigid against the central force caused by the rotation of the helicopter propeller, and flexibility against the blade's flapping and rotational movement. In this study, the Group Method of Data Handling (GMDH) model is used to estimate the maximum stresses in elastomeric layers in a spherical elastomeric bearing under compression and angular displacement loading. ? (angular displacement loading), P (pressure loading), a (radius of axis), ß0 (first joint angle), cos (ß0) (first joint angle cosine value), ß1 (second joint angle), ß2 (third joint angle), ft (cone angle), fp (angle between the direction of the pressure loading and the plane perpendicular to the elastomer layer), cos (fp), D (elastomer layer outer diameter), ne (elastomer layer number), d (elastomer layer hole diameter) and H (elastomer layer thickness) were used as input features for GMDH model. The results obtained with GMDH were also compared with different machine learning methods such as ANN, SVM, RF. According to the results obtained, GMDH model was found to be more successful than other models in estimating spherical elastomeric bearing. [ABSTRACT FROM AUTHOR]

Let G n , k D be the set of all k -connected graphs of order n with diameter D. In this paper, we determine the graphs with maximal signless Laplacian spectral radius among all graphs in G n , k D. [ABSTRACT FROM AUTHOR]

Let G be a connected irregular graph on n vertices with maximum degree Δ and diameter D. The spectral radius of G, which is denoted by ρ (G) , is the largest eigenvalue of the adjacency matrix of G. In this paper, we study the lower bound of Δ - ρ (G) . As a result, a new lower bound is obtained which improves the known lower bounds of Δ - ρ (G) . [ABSTRACT FROM AUTHOR]

The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to Dirac (1952), Ore (1960), Jung (1978), and Nash-Williams (1971). The idea was to show that the global concept of Hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. (The ball of radius r centered at a vertex u is a subgraph of G induced by the set of vertices whose distances from u do not exceed r.) Such results are called localization theorems and present a possibility to extend known classes of finite Hamiltonian graphs. In this paper we formulate a general approach for finding localization theorems and use this approach to formulate local analogues of well-known results of Bauer et al. (1989), Bondy (1980), Häggkvist and Nicoghossian (1981), and Moon and Moser (1963). Finally we extend two of our results to infinite locally finite graphs and show that they guarantee the existence of Hamiltonian curves, introduced by Kündgen, Li and Thomassen (2017). [ABSTRACT FROM AUTHOR]

Copyright of China Mechanical Engineering is the property of Editorial Board of China Mechanical Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let S (m, d, k) be the set of k-uniform supertrees with m edges and diameter d, and S1 (m, d, k) be the k-uniform supertree obtained from a loose path u1, e1, u2, e2, ..., ud, ed, ud+1 with length d by attaching m–d edges at vertex u⌊d/2⌋+1. In this paper, we mainly determine S1 (m, d, k) with the largest signless Laplacian spectral radius in S (m, d, k) for 3 ⩽ d ⩽ m − 1. We also determine the supertree with the second largest signless Laplacian spectral radius in S (m, 3, k). Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices). [ABSTRACT FROM AUTHOR]

For 0 ≤ α ≤ 1, Nikiforov proposed to study the spectral properties of the family of matrices Aα(G) = αD(G)+(1 - α)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The α-spectral radius of G is the largest eigenvalue of Aα(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the α-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the α-spectral radius for unicyclic graphs G with maximum degree ∆ ≥ 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest α-spectral radius among trees, and the unique tree with the largest α-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the α-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively. [ABSTRACT FROM AUTHOR]

A graph G = (V , E) is distance hereditary if every induced path of G is a shortest path. In this paper, we show that the eccentricity function e (v) = max ⁡ { d (v , u) : u ∈ V } in any distance-hereditary graph G is almost unimodal, that is, every vertex v with e (v) > r a d (G) + 1 has a neighbor with smaller eccentricity. Here, r a d (G) = min ⁡ { e (v) : v ∈ V } is the radius of graph G. Moreover, we use this result to fully characterize the centers of distance-hereditary graphs. Several bounds on the eccentricity of a vertex with respect to its distance to the center of G or to the ends of a diametral path are established. Finally, we propose a new linear time algorithm to compute all eccentricities in a distance-hereditary graph. [ABSTRACT FROM AUTHOR]

We show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time O (n · k + ⌈ log n ⌉ k · 2 k log n) , where k is linear in the treewidth of the graph. For every ϵ > 0 , this bound is n 1 + ϵ exp O (k) , which matches a hardness result of Abboud et al. (in: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2016. 10.1137/1.9781611974331.ch28) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comput Geom 42:815–824, 2009. 10.1016/j.comgeo.2009.02.001) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log d n to d + ⌈ log n ⌉ d , as originally observed by Monier (J Algorithms 1:60–74, 1980. 10.1016/0196-6774(80)90005-X). We also investigate the parameterization by vertex cover number. [ABSTRACT FROM AUTHOR]

In this paper, we determine the unique graph among complements of unicyclic graphs which maximizes the distance spectral radius. We also characterize the unique graph among complements of unicyclic graphs of diameter three which maximizes the least distance eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper we show how the α -spectral radius changes under the edge grafting operations on connected k -uniform hypergraphs. We characterize the extremal hypertree for α -spectral radius among k -uniform non-caterpillar hypergraphs with given order, size and diameter. We also characterize the supertree with the second largest α -spectral radius among all k -uniform supertrees on n vertices. [ABSTRACT FROM AUTHOR]

Surface topography, as one of the significant roles in surface integrity, has a great impact on the performances and service life of the machined parts. This research focuses on the surface topography model for the ball-nose end miller in machining of AISI P20 steel. First, the model is developed to predict the surface topography and surface roughness in ball-nose end milling process. Secondly, the accuracy of the developed surface topography model was verified by a series of milling experiments. Thirdly, the effects of corner radius and diameter of ball-nose end miller on surface roughness is analyzed, it is observed that the ratio of feed per tooth (fz) to radial depth of cutting (ae) for obtaining minimum surface roughness is related to the ratio of diameter (D) to corner radius (r) of ball-nose end miller. Finally, in terms of the minimum surface roughness, a mathematical model is established with consideration of corner radius and diameter of ball-nose end miller. This research indicates that proper selection of cutting parameters (fz and ae) with consideration of diameter and radius corner of ball-nose end miller is a novel avenue for acquiring desired surface roughness. [ABSTRACT FROM AUTHOR]

For 0 ≤ α < 1 and a uniform hypergraph G, we consider the spectral radius of A α (G) = α D (G) + (1 − α) A (G) , which is called the α-spectral radius of G, where D (G) and A (G) are the diagonal tensor of degrees and the adjacency tensor of G, respectively. We give a upper bound for the α-spectral radius of an n-vertex connected irregular k-uniform hypergraph with 2 ≤ k < n using number of vertices, maximum degree and diameter. [ABSTRACT FROM AUTHOR]

We investigate the L 1 geodesic farthest neighbors in a simple polygon P, and address several fundamental problems related to farthest neighbors. Given a subset S ⊆ P , an L 1 geodesic farthest neighbor of p ∈ P from S is one that maximizes the length of L 1 shortest path from p in P. Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the L 1 geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset S ⊆ P with respect to the L 1 geodesic distance after removing redundancy. [ABSTRACT FROM AUTHOR]

It is important for the thin-walled tube with relative bending radius of 1 to be bent with mandrel in order to avoid the defects such as wrinkling and cross-section distortion. In this paper, a novel push-bending method using sectional elastomers as mandrel was proposed to form the 5A02 thin-walled aluminum alloy tube with relative bending radius of 1 and 90° bending angle. Cylindrical polyurethane rubber (CPR) with shore hardness of 80A was chosen as the sectional elastomers. The influence of diameter (d), thickness (t), total length (L), and arrangement of CPR filler on the push-bending deformation of tube was first explored by means of simulations and experiments. According to the simulation results, 5A02 thin-walled aluminum alloy tube with relative bending radius of 1 was successfully fabricated with the sectional CPR as mandrel in the self-developed equipment of push-bending. The experimental results were in good agreement with the simulation results. The optimum parameters of CPR filler in push-bending process were obtained as follows: 37 mm diameter, a piece of CPR with 20 mm thickness contacted with the punch, and other 12 pieces of CPR with 10 mm thickness. [ABSTRACT FROM AUTHOR]

• Delamination of the REBCO layer started to form exclusively on the scratch location at tube diameter of 7 mm while the rest of the REBCO surface started to delaminate at tube diameter of 5 mm. • The scratches on the CC tape surface increase the critical bending radius when tape bent (in our case by 2 mm). • The critical current degradation started to occur when first signs of REBCO delamination appeared. • Scratches induce REBCO delamination during the bending process resulting in premature superconductivity degradation of the CC tape. • The presence of scratch line seems to be a crucial factor for REBCO delamination mostly for CC tapes with the thin overlayer(s). The aim of this work is to study delamination promotion of the coated conductor (CC) tape under the influences of the scratches on the Ag surface when bending. For this purpose, the microstructural and superconducting properties of the CC tape bent in helical fashion with different former diameters were analyzed. Such CC tape that exhibits visible damage along its length in the form of "scratch line" caused on the final product was used. The CC tape was bent at a constant lay angle of 45° with a superconducting layer positioned inward to the former core experiencing compression. Transport current measurements in liquid nitrogen were performed to study superconducting properties of the CC tape. The results showed that delamination of the REBCO (Rare-Earth Barium Copper Oxide) layer started to form exclusively on the scratch location at tube diameter of 7 mm (corresponding to strain ε = −0.71 %) while the rest of the REBCO surface started to delaminate at tube diameter of 5 mm (ε = −0.99 %). As the superconducting properties of the CC tape are related with the REBCO delamination, the critical current degraded when first signs of REBCO delamination appeared as expected. Thus, it was observed that the scratches induce REBCO delamination during the bending process resulting in increase of critical bending radius. [ABSTRACT FROM AUTHOR]

The results from the foil liner implosion on a 2-MA generator are reported. To sharpen the front of the current through the liner up to several nanoseconds, the liner area is prefilled with plasma with a density of 1016–1017 cm−3. The current switches to the liner in the process of sweeping the injected plasma by J × B force. The nanosecond rise time of the current through the liner makes it possible to reduce its initial radius to less than 1 mm and use available foil to fabricate the liner. In the experiment, liners with initial diameters of 0.65–2.0 mm made out of 8-μm-thick aluminum foil were imploded in 6–27 ns. [ABSTRACT FROM AUTHOR]

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the manifold and of the corresponding sphere is likewise bounded. These results are actually obtained in the more general context of (possibly nonsmooth) metric measure spaces with curvature‐dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

The aim is to provide a necessary and sufficient condition under which both a graph and its complement have (a) identical radius, (b) identical diameter, (c) identical center, and (d) identical periphery. [ABSTRACT FROM AUTHOR]

Let ζ be the set of all cliques in a connected graph G and C,C′ ∈ ζ. A clique- to-clique C - C′ path P is a u - v path, where u ∈ C and v ∈ C′, in which P contains no vertices of C and C′ other than u and v. The clique-to-clique detour distance, D(C,C′) is the length of a longest C - C′ path. A C - C′ path of length D(C,C′) is called a C - C′ detour. The clique-to-clique detour eccentricity eD3(C) of a clique C in G is the maximum clique-to-clique detour distance from C to a clique C′ ∈ ζ in G. The clique-to-clique detour radius R3 of G is the minimum clique-to-clique detour eccentricity among the cliques of G, while the clique-to-clique detour diameter D3 of G is the maximum clique-to-clique detour eccentricity among the cliques of G. It is shown that R3 ≤ D3 for every connected graph G and that every two positive integers a and b with 2 ≤ a ≤ b are realizable as the clique-to- clique detour radius and the clique-to-clique detour diameter of some connected graph. Also it is shown that for any two positive integers a and b with 2 ≤ a ≤ b are realizable as the clique-to-clique radius (diameter) and the clique-to-clique detour radius (diameter) of some connected graph. [ABSTRACT FROM AUTHOR]

Copyright of China Sciencepaper is the property of China Sciencepaper and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Ostrand posed the following two questions in 1973. (1) What is the maximum girth of a graph with radius r and diameter d ? (2) What is the minimum circumference of a graph with radius r and diameter d ? Question 2 has been answered by Hrnčiar who proves that if d ≤ 2 r − 2 the minimum circumference is 4 r − 2 d . In this note we first answer Question 1 by proving that the maximum girth is 2 r + 1 . This improves on the obvious upper bound 2 d + 1 and implies that every Moore graph is self-centered. We then prove a property of the blocks of a graph which implies Hrnčiar’s result. [ABSTRACT FROM AUTHOR]

We determine the trees with the minimal Laplacian spectral radius with diameter 4, which settles a conjecture posed by Liu et al. (2009) [5] . [ABSTRACT FROM AUTHOR]

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. P. Hansen and D. Stevanović (2008) [9] determined the graphs with maximum spectral radius among all connected graphs of order n with diameter D . In this paper, we generalize this result to k -connected graphs of order n with diameter D . [ABSTRACT FROM AUTHOR]

A bug Bugp,r1,r2 is a graph obtained from a complete graph Kp by deleting an edge uv and attaching the paths Pr1 and Pr2 by one of their end vertices at u and v, respectively. Let Q(G) be the signless Laplacian matrix of a graph G and q1(G) be the spectral radius of Q(G). It is known that the bug B0 = Bugn-d+2,[d/2],[d/2] maximizes q1(G) among all graphs G of order n and diameter d. For a bug B of order n and diameter d, n - d is an eigenvalue of Q(B) with multiplicity n - d - 1. In this paper, we prove that remainder d + 1 eigenvalues of Q(B), among them q1(B), can be computed as the eigenvalues of a symmetric tridiagonal matrix of order d+1. Finally, we show that q1(B0) can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order d/2 + 1 whenever d is even. [ABSTRACT FROM AUTHOR]

The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G , defined as D Q ( G ) = T r ( G ) + D ( G ) , where D ( G ) is the distance matrix of G and T r ( G ) is the diagonal matrix of vertex transmissions of G . In this paper we determine upper and lower bounds on the minimal and maximal entries of the principal eigenvector of D Q ( G ) and characterize the extremal graphs. In addition, we obtain a lower bound on the distance signless Laplacian spectral radius of G based on its order and independence number, and characterize the extremal graph. [ABSTRACT FROM AUTHOR]

Given a metric $$(V,d)$$ and an integer $$k$$ , we consider the problem of partitioning the points of $$V$$ into at most $$k$$ clusters so as to minimize the sum of radii or the sum of diameters of these clusters. The former problem is called the minimum sum of radii (MSR) problem and the latter is the minimum sum of diameters (MSD) problem. The current best polynomial time algorithms for these problems have approximation ratios 3.504 and 7.008, respectively. We call a cluster containing a single point, a singleton cluster. For the MSR problem when singleton clusters are not allowed, we give an exact algorithm for metrics induced by unweighted graphs. In addition, we show that in this case, a solution consisting of the best single cluster for each connected component of the graph is a $$\frac{3}{2}$$ -approximation algorithm. For the MSD problem on the plane with Euclidean distances, we present a polynomial time approximation scheme. In addition, we settle the open problem of complexity of the MSD problem with constant $$k$$ by giving a polynomial time exact algorithm in this case. The previously best known approximation algorithms for MSD on the plane or for MSD with constant $$k$$ have both ratio 2. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n ≥ 3 . The remoteness ρ = ρ ( G ) is the maximum, over all vertices, of the average distance from a vertex to all others. The radius r = r ( G ) is the minimum, over all vertices, of the eccentricity of a vertex. Aouchiche and Hansen (2011) conjectured that ρ − r ≥ 3 − n 4 if n is odd and ρ − r ≥ 2 n − n 2 4 ( n − 1 ) if n is even. In this paper, we confirm this conjecture. In addition, we completely characterize extremal graphs attaining the lower bound. [ABSTRACT FROM AUTHOR]

The oriented diameter of a bridgeless graph G is min{diam(H) | H is a strang orientation of G}. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and η(G), where rad(G) is the radius of G and η (G) is the smallest integer number such that every edge of G is contained in a cycle of length at most η (G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G. [ABSTRACT FROM AUTHOR]

We consider the construction of sparse covers for planar graphs and other graphs that exclude a fixed minor. We present an algorithm that gives a cover for the γ-neighborhood of each node. For planar graphs, the cover has radius less than 16 γ and degree no more than 18. For every n node graph that excludes a minor of a fixed size, we present an algorithm that yields a cover with radius no more than 4 γ and degree O(log n). This is a significant improvement over previous results for planar graphs and for graphs excluding a fixed minor; in order to obtain clusters with radius O( γ), it was required to have the degree polynomial in n. Our algorithms are based on a recursive application of a basic routine called shortest-path clustering, which seems to be a novel approach to the construction of sparse covers. Since sparse covers have many applications in distributed computing, including compact routing, distributed directories, synchronizers, and Universal TSP, our improved cover construction results in improved algorithms for all these problems, for the class of graphs that exclude a fixed minor. [ABSTRACT FROM AUTHOR]

A diameter graph in $${\mathbb R}^d$$ is a graph whose set of vertices is a finite subset of $${\mathbb R}^d$$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of different extremal properties of diameter graphs in $${\mathbb R}^4$$ and on a three-dimensional sphere. We prove an analog of Vázsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than $$1/\sqrt{2}$$ . We prove Schur's conjecture for diameter graphs in $${\mathbb R}^4.$$ We also establish the maximum number of triangles a diameter graph in $${\mathbb R}^4$$ can have, showing that the extremum is attained only on specific Lenz configurations. [ABSTRACT FROM AUTHOR]

In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $. [ABSTRACT FROM PUBLISHER]

Highlights: [•] X-ray nanotomography demonstrated including meshing a nanofiber surface in 3D. [•] Developed processing of reconstructions for interpretation of 3D nano-geometry. [•] Nanotomography of the 220nm diameter fiber showed 19% local deviation in radius. [ABSTRACT FROM AUTHOR]

The rainbow connection number, rc( G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc( G) ≤ r( r + 2). We demonstrate that this bound is the best possible for rc( G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc( G) can be arbitrarily larger than its radius ( K for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc( G) ≤ rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4 n/5 − 1, where n is order of the graph (Caro et al. in Electron J Comb 15(1):Research paper 57, 13, ). It is known that computing rc( G) is NP-Hard (Chakraborty and fischer in J Comb Optim 1-18, ). Here, we present a ( r + 3)-factor approximation algorithm which runs in O( nm) time and a ( d + 3)-factor approximation algorithm which runs in O( dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r. [ABSTRACT FROM AUTHOR]