Narrow Results

The topological index is just one of several very useful tools that graph theory has made available to chemists. Topological indices are invariants of real numbers under graph isomorphisms. Several topological indices have been defined. Some of them are used to model chemical, pharmaceutical and other properties of molecules. The eccentric connectivity index (eci) is also a topological index. The eci of $G$, denoted by ${𝜀}^c(G)$, is defined as ${𝜀}^c(G)={\sum }_{v\in V(G)}deg(v)e(v)$, where $deg(v)$ represents the degree of a vertex $v$ and $e(v)$ is its eccentricity. In this paper, exact formula for the eci of complementary prisms is derived.

We review recent results from the RHIC beam energy scan (BES) program, aimed to study the Quantum Chromodynamics (QCD) phase diagram. The main goals are to search for the possible phase boundary, softening of equation of state or first order phase transition, and possible critical point. Phase-I of the BES program has recently concluded with data collection for Au+Au collisions at center-of-mass energies of 7.7, 11.5, 19.6, 27 and 39 GeV. Several interesting results are observed for these lower energies where the net-baryon density is high at the mid-rapidity. These results indicate that the matter formed at lower energies (7.7 and 11.5 GeV) is hadron dominated and might not have undergone a phase transition. In addition, a centrality dependence of freeze-out parameters is observed for the first time at lower energies, slope of directed flow for (net)-protons measured versus rapidity shows an interesting behavior at lower energies, and higher moments of net-proton show deviation from Skellam expectations at lower energies. An outlook for the future BES Phase-II program is presented and efforts for the detailed study of QCD phase diagram are discussed.

This paper considers the discontinuous characteristics of a real aero-engine rotor system, that is, the existence of bolted connection characteristics, and establishes a new bolted connection rotor system model. Taking into account the bending stiffness and the nonlinear Hertzian contact force of the rolling bearing, the Newmark-$\beta$ numerical method is used to solve the system response, and the influence of the bending stiffness on the system is studied. Moreover, the effects of bending stiffness and eccentricity on the system dynamics are analyzed. The results show that the nonlinear phenomena of the system are more abundant and the critical speed of the system is higher when the bending stiffness is involved. With the increase of bending stiffness, the critical speed of the system increases, and the frequency component of the system becomes more complex. Then, the influence of eccentricity on the system is studied based on the bending stiffness. It is found that the greater the eccentricity, the greater the critical speed of the rotor and the greater the amplitude of the main frequency. In the case of the same eccentricity, the main frequency increases as the rotational speed increases, and the frequency doubling component appears in the 2-period motion. This paper provides a basis for predicting the nonlinear response of bolted rotor-bearing system.

The Euclidean centre (centre of the smallest enclosing sphere) of a set of points P in two or more dimensions is unstable; small perturbations at only a few points of P can result in an arbitrarily large relative change in the position of the Euclidean centre. Any centre function more stable than the Euclidean centre is eccentric; that is, its associated radius exceeds the radius of the smallest enclosing circle for some point sets P. Motivated by applications in mobile facility location (in which clients move continuously with some maximum velocity) we seek alternative notions of centrality that are stable while maintaining low eccentricity. In general there is a trade-off; centre functions with lower eccentricity are less stable. In an attempt to balance the conflicting goals of closeness of approximation and stability, we apply the Steiner centre, traditionally defined for convex polytopes, as a centre function of a set of points in the plane. Although previously defined, the notion of a Steiner centre had not been analyzed in terms of its approximation of the Euclidean centre. Exploiting the equivalence of the two definitions of the Steiner centre established by Shephard,²⁷ we prove the stability of the Steiner centre is π/4 and show that the associated radius is at most 1.1153 times the Euclidean radius of any point set P. It follows that a mobile facility located at the Steiner centre of the positions of a set of mobile clients remains close to the Euclidean centre of the clients yet never moves with relative velocity that exceeds 4/π.

In high energy heavy-ion collisions, the final anisotropic flow coefficients and their corresponding event–plane correlations are considered as the medium evolutional response to the initial geometrical eccentricities and their corresponding participant–plane correlations. We formulate a systematic theoretical analysis to study the hydrodynamical responses concerning higher-order effects in $\mathrm{\text{Pb}}+\mathrm{\text{Pb}}$ collisions at $\sqrt{s_{\mathrm{\text{NN}}}}=2.76$TeV by using Monte Carlo (MC) Glauber model. To further understand the transformations of the initial participant–plane correlation, we construct a new set of events which randomize the directions of the initial participant–planes of the original events. Our results indicate that the final strong event–plane correlations are mainly transformed from the large initial eccentricities, rather than the strong participant–plane correlations. However, the large flow coefficients and the discrepancies between the flow coefficients calculated by the single-shot and event-by-event simulations in peripheral collisions are relevant to those strong initial participant–plane correlations.

The exact deg-centric graph of a simple, connected graph $G$, denoted by $G_{\mathrm{\text{ed}}}$, is a graph constructed from $G$ such that $V(G_{\mathrm{\text{ed}}})=V(G)$ and $E(G_{\mathrm{\text{ed}}})=\{v_i{v}_j:d_G(v_i,v_j)={\mathrm{\text{deg}}}_G(v_i)\}$. In this paper, the concepts of exact deg-centric graphs and iterated exact deg-centrication of a graph are introduced and discussed.

The loads induced on the spacecraft orbiting the Earth by the deploying elastic arm are investigated. The coupled equations of motion of the arm with the vehicle orbital mechanics are used to describe the 3D dynamic behavior of the flexible-appendage and the related disturbing loads induced on the spacecraft. To this end, an equivalent dynamical system is derived for the arm by applying an attached Non-Newtonian Reference Frame which is subjected to the orbital motion and geocentric pointing maneuver of the spacecraft. With the help of the Assumed Modes Method, the behavior of the arm attached to the spacecraft in Keplerian orbits is studied. The results show that deploying the arm in some specific directions relative to the orbital plane leads to serious coupling between two lateral displacements. In addition, the effects of specific orbital parameters on arm responses and resulting induced loads are studied for the cases of "True Anomaly of spacecraft at deployment time, and Eccentricity of elliptical orbits". The prediction of disturbing loads induced on spacecraft helps design the robust attitude control system. Further, the positioning accuracy of the payloads (installed on the arm-tip) can be estimated by employing the obtained arm responses in the orbital motion, which enables us to determine the undesirable motions and predict any required control system for the arm.

This paper is concerned with the lateral and torsional coupled vibration of monosymmetric I-beams under moving loads. To this end, a train is modeled as two subsystems of eccentric wheel loads of constant intervals to account for the front and rear wheels. By assuming the lateral and torsional displacements to be restrained at the two ends of the beam, both the lateral and torsional displacements are approximated by a series of sine functions. The method of variation of constants is adopted to derive the closed-form solution. For the most severe condition when the last wheel load is acting on the beam, both the conditions of resonance and cancellation are identified. Once the condition of cancellation is enforced, the resonance response can always be suppressed, which represents the optimal design for the beam. Since the condition for suppressing the torsional resonance is exactly the same as that for the vertical resonance, this offers a great advantage in the design of monosymmetric I-beams, as no distinction needs to be made between the suppression of vertical or torsional resonance.

This paper presents a dynamical model of a fluid-conveying pipe spinning about an eccentric axis. The coupled bi-flexural–torsional free vibration and stability are analyzed for such a doubly gyroscopic system. The partial differential equations of motions are derived by the extended Hamilton principle, and are then truncated by a 4-term Galerkin technique. The frequency and energy evolutions and representative mode shapes in the two transverse directions and torsional direction are investigated to unveil the essential dynamical attributes of the system. It is indicated that the stability of the present system mainly depends on spinning speed, flow velocity and eccentricity, while the torsional frequencies are almost immune to the flow velocity. The pipe reveals ‘traveling-wave’ modal vibrations in both transverse directions, and a general ‘standing-wave’ modal vibration in the torsional direction.

There are several critical factors which have adverse effects on the workpiece surface finish, dimensional accuracy, tool life, etc. Eccentricity is one of them which can be manipulated when the spindle, tool holder and milling cutter revolve around the line parallel to their cutting axes and crop up due to clamping deviation and geometric imperfections. This work was driven by the observation that there should an approach through which final eccentric error can be reduced as other researchers only displayed the effects of eccentricity on cutting forces, etc. This task was carried out through performing numerous experiments on the CNC milling center and measuring geometric imperfections from the whole body of machining elements at different orientations with dial gauge to find out whether different orientations of elements have effects on the eccentricity. If they have noticeable effects then such orientations of elements exhibiting the minimum eccentricity can be selected which in turn may improve the accuracy and precision of workpieces and reduce the final cutting edge error. Our observation showed that orientations have remarkable effects on the eccentricity pretending lowest final eccentric error which is unable to be ignored.

The application of graph theory in structural biology offers an alternative means of studying 3D models of large macromolecules such as proteins. The radius of gyration, which scales with exponent $\sim 0.4$, provides quantitative information about the compactness of the protein structure. In this study, we combine two proven methods, the graph-theoretical and the fundamental scaling laws, to study 3D protein models. This study shows that the mean node degree (MND) of the protein graphs, which scales with exponent 0.038, is scale-invariant. In addition, proteins that differ in size have a highly similar node degree distribution. Linear regression analysis showed that the graph parameters (radius, diameter, and mean eccentricity) can explain up to 90% of the total radius of gyration variance. Thus, the graph parameters of radius, diameter, and mean eccentricity scale match along with the same exponent as the radius of gyration. The main advantage of graph eccentricity compared to the radius of gyration is that it can be used to analyze the distribution of the central and peripheral amino acids/nodes of the macromolecular structure.

The unsteady viscous flow around a 12000TEU ship model entering the Third Set of Panama Locks with different eccentricity is simulated by solving the unsteady Reynolds averaged Navier–Stokes (RANS) equations in combination with the $k-\omega$SST turbulence model. Overset grid technology is utilized to maintain grid orthogonality and the effects of the free surface are taken into account. The hydrodynamic forces, vertical displacement as well as surface pressure distribution are predicted and analyzed. First, a benchmark test case is designed to validate the capability of the present methods in the prediction of the viscous flow around the ship when maneuvering into the lock. The accumulation of water in front of the ship during entry into a lock is noticed. A set of systematic computations with different eccentricity are then carried out to examine the effect of eccentricity on the ship–lock hydrodynamic interaction.

In this study, the anti-cavitation performance and cavitation flow characteristics in a hydrodynamic levitated micropump were investigated based on numerical simulation and experiment. The cavitation characteristic curves and the development process of cavitation in the levitated micropump was firstly analyzed. Special emphasis was put on the effects of eccentricity on the anti-cavitation performance. The results show that as the eccentricity increases, the critical cavitation number gradually decreases, indicating that the eccentric rotation is beneficial to improve the anti-cavitation ability of the levitated micropump. The coupling effects between the radial force on the impeller and cavitation were also numerically studied. With the decrease of cavitation number, the radial force on the impeller gradually declines at first, then has a sudden increase and finally reduces with fluctuation. The drop of the radial force will lead to the decrease of eccentricity, resulting in the deterioration of cavitation further. In addition, the unsteady pressure pulsation was analyzed. The predominant frequencies of pressure pulsation are the blade passing frequency (BPF) and the harmonic frequency of BPF under both noncavitation and critical cavitation. Under critical cavitation, the amplitude of BPF has a drop, while the amplitude of low frequency less than BPF becomes larger.

In this research, energy absorption characteristics of nested thin-walled aluminum structures with square cross-section have been investigated, and the effect of the three parameters: eccentricity, rotation, and tube aspect ratio on the energy absorption characteristics of the structure has been studied. The properties investigated were the specific energy absorption and crush force efficiency. An aluminum 1050 sheet was used to build the tubes. After making the samples and determining how the tubes were placed with each other, the structures were subjected to quasi-static axial loading. Also, for parametric investigations, the behavior of the structures was simulated using LS-Dyna software. The results showed that the simultaneous effect of the three desired parameters has increased the specific absorbed energy and the crushing force efficiency. When the tubes were at maximum eccentricity, the specific energy absorption and crushing force efficiency increased by reducing the aspect ratio and rotation angle. On average, the best sample’s specific energy absorption and crush force efficiency increased by 41.9% and 22.2%, respectively, compared to the single tube.

We generalize the definition of a fuzzy graph by replacing minimum in the basic definitions with an arbitrary $t$-norm. The reason for this is that some applications are better modeled with a $t$-norm other than minimum. We develop a measure on the susceptibility of trafficking in persons for networks by using a $t$-norm other than minimum. We also develop a connectivity index for a fuzzy network. In one application, a high connectivity index means a high susceptibility to trafficking. In the other application, we use a method called the eccentricity of an origin country to determine the susceptibility of a network to trafficking in persons. The models rest on the vulnerabilities and the government responses of countries to trafficking.

We introduce the notion of a vague incidence graph and its eccentricity. We apply the results to problems involving human trafficking and illegal immigration. We are particularly interested in the roll played by countries’ vulnerability and their government’s response to human trafficking. We show of the leading illegal immigration routes through Mexico to the United States that Somalia has the highest eccentricity. We also provide measures on how much a region needs to reduce flow or increase government response to be modelled by a fuzzy incidence graph. It turns out that the Commonwealth of Independent States has the largest measure for both vulnerability and government response than any other region.

The Wiener number $W(G)$ of a graph $G$ was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by $WW(G)$ and defined as $WW(G)=\frac{1}{2}{\sum }_{u\in V}[d(u|G)+d^2(u|G)]$. In this paper, we establish some upper and lower bounds for $WW(G)$, in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.

Let $G$ be a simple connected graph with vertex set $V_G$ and edge set $E_G$. The eccentricity ${\mathrm{\text{ec}}}_G(x)$ of a vertex $x$ in $G$ is the largest distance between $x$ and any other vertex of $G$. The eccentric adjacency index (also known as Ediz eccentric connectivity index) of $G$ is defined as ${\xi }^{\mathrm{\text{ad}}}(G)={\sum }_{x\in V_G}\frac{S_G(x)}{{\mathrm{\text{ec}}}_G(x)}$, where $S_G(x)$ is the sum of degrees of neighbors of the vertex $x\in V_G$. In this paper, we determine the unicyclic graphs with largest eccentric adjacency index among all $n$-vertex unicyclic graphs with a given girth. In addition, we find the tree with largest eccentric adjacency index among all the $n$-vertex trees with a fixed diameter.

For $a,b\in ℝ$, we define the general eccentric distance sum of a connected graph $G$ as ${\mathrm{\text{EDS}}}_{a,b}(G)={\sum }_{v\in V(G)}{(ec{c}_G(v))}^a{(D_G(v))}^b$, where $V(G)$ is the vertex set of $G$, $ec{c}_G(v)$ is the eccentricity of a vertex $v$ in $G$, $D_G(v)={\sum }_{w\in V(G)}{d}_G(v,w)$ and $d_G(v,w)$ is the distance between vertices $v$ and $w$ in $G$. This index generalizes several other indices of graphs. We present some bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees of given order, graphs of given order and vertex connectivity and graphs of given order and number of pendant vertices. The extremal graphs are presented as well.

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 $a\in V$ is considered as the maximum length of all shortest paths starts from $a$ to the remaining nodes, i.e., $e(a)=max\{d(a,u_1):u_1\in V\}$. 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., $d(G)=max\{e(a):a\in V\}$. 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., $r(G)=min\{e(a):a\in V\}$. 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., $c(G)=\{x\in V:e(x)=r(G)\}$. 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.