Total Eccentricity Research Articles

On the Eccentric Complexity of Graphs

Let G be a simple connected graph. The eccentric complexity of graph G is introduced as the number of different eccentricities of its vertices. A graph with eccentric complexity equal to one is called self-centered. In this paper, we study the eccentric complexity of graph under several graph operations such as complement of graph, line graph, Cartesian product, sum, disjunction and symmetric difference of graphs. Also, we present an infinite family of non-vertex-transitive self-centered graphs and we prove that all such graphs are 2-connected. Further, for any D and k where $$k \le \frac{D-1}{2}$$ , we construct an infinite family of non-vertex-transitive graphs with eccentric complexity k and diameter D. Extremal graphs with minimum or maximum total eccentricity among all graphs with given eccentric complexity are determined. We also consider a family of nanotubes and show that it is extremal with respect to the eccentric complexity among all fullerene graphs. At the end we also indicate some possible directions of further research.

The Total Eccentricity Sum of Non-adjacent Vertex Pairs in Graphs

Classical topological indices, such as Zagreb indices ( $$M_{1}$$ and $$M_2$$ ) and the well-studied eccentric connectivity index ( $$\xi ^{c}$$ ) directly or indirectly consider the total contribution of all edges in a graph. By considering the total degree sum of all non-adjacent vertex pairs in a graph, Ashrafi et al. (Discrete Appl Math 158:1571–1578, 2010) proposed two new Zagreb-type indices, namely the first Zagreb coindex ( $$\overline{M}_{1}$$ ) and second Zagreb coindex ( $$\overline{M}_{2}$$ ), respectively. Motivated by Ashrafi et al., we consider the total eccentricity sum of all non-adjacent vertex pairs, which we call the eccentric connectivity coindex ( $$\overline{\xi }^{c}$$ ), of a connected graph. In this paper, we study the extremal problems of $$\overline{\xi }^{c}$$ for connected graphs of given order, connected graphs of given order and size, and the trees, unicyclic graphs, bipartite graphs containing cycles and triangle-free graphs of given order, respectively. Additionally, we establish various lower bounds for $$\overline{\xi }^{c}$$ in terms of several other graph parameters.

Eccentricity sums in trees

The eccentricity of a vertex, eccT(v)=maxu∈TdT(v,u), was one of the first, distance-based, tree invariants studied. The total eccentricity of a tree, Ecc(T), is the sum of the eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios Ecc(T)/eccT(u), Ecc(T)/eccT(v), eccT(u)/eccT(v), and eccT(u)/eccT(w) where u,w are leaves of T and v is in the center of T. In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence.

Total eccentricity index of some composite graphs

The total eccentricity index of a graph G is the sum of eccentricities of all the vertices of G. In this paper, we first derive some sharp upper and lower bounds of total eccentricity index of different subdivision graphs and then determine some explicit expression of the total eccentricity index of the double graph, extended double cover graph and some generalized thorn graphs.

图运算下的总离心率及多项式

让G表示一个简单连通图，图G的总离心率及多项式分别定义为，，这里是顶点ν在G中的离心率。在本文中，计算了在图运算下图的双覆盖图和拓展双覆盖图以及剖分图的总离心率及多项式，并给出具体的关系表达式和一些界。 Let G be a simple connected graph. The total eccentricity and total eccentricity polynomial of a graph G are defined as and , where denotes the eccentricity of vertex ν in G. In this paper, the total eccentricity and total eccentricity polynomial of double cover graph and extended double cover graph and subdivision graph of a given graph under the graph operations are computed and the exact expressions and some bounds are given.

Total Eccentricity Index of the Generalized Hierarchical Product of Graphs

Let \(G=(V(G),E(G))\) be a connected graph. The total eccentricity index of \(G\) is defined as \(\zeta (G)=\sum \nolimits _{v\in V(G)}{{{\varepsilon }_{G}}(v)}\), where \({{\varepsilon }_{G}}(v)\) is the eccentricity of the vertex \(v\). In this paper, we compute the total eccentricity of generalized hierarchical product of graphs. Moreover, we derive some explicit formulae for total eccentricity index of F-sum graph, Cartesian product, Cluster product and Corona product of graphs and apply those results to find the total eccentricity index of various classes of chemical graphs and nanostructures.

Output Feedback Control of a Dual‐Excenter Vibration Actuator via qLPV Model and TP Model Transformation

AbstractVibration actuators are widely used in handheld devices to provide vibrotactile feedback or silent notification to the users. In most cases, miniature DC motors with eccentric rotors or the so‐called coin‐type shaftless vibration motors are utilized. The common disadvantage of the single rotor designs is that the frequency and the intensity of the generated vibration cannot be adjusted separately. The construction composed of two independently driven coaxial eccentric rotors – which makes a strongly coupled nonlinear system – allows for the separate control of the frequency and amplitude by the adjustment of the angular speed and the total eccentricity. This paper presents a complete control design approach based on quasi‐linear parameter varying (qLPV) modelling and linear matrix inequality (LMI)‐based synthesis utilizing the tensor product (TP) model transformation to determine the convex polytopic representation of the parameter dependent nonlinear system. The design approach is demonstrated via a concrete numerical example using the parameters of a real dual‐excenter prototype device. The control performance is validated through numerical simulations. This case study goes through a complete nonlinear control problem from the modelling phase to the implementation‐ready controller, while drawing generalizable conclusions for qLPV modelling.

Total Eccentricity of some Graph Operations

Let G be a simple connected graph. The total eccentricity of a graph G,ζ(G), is defined as ζ(G)=∑v∈V(G)ecG(v), where ecG(v) of a vertex v∈V(G) is the maximum distance between v and any other vertex in G. In this paper, the total eccentricity of some graph operations are computed and then a bound for that of tensor product is presented.

Relationship Between Augmented Eccentric connectivity index and Some Other Graph Invariants

The augmented eccentric connectivity index of a graph which is a generalization of eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph. In this paper we established some relationships between augmented eccentric connectivity index and several other graph invariants like number of vertices, number of edges, maximum vertex degree, minimum vertex degree, the total eccentricity index the first Zagreb indices and the second multiplicative Zagreb index.

Discovery of a double eclipsing binary with periods near a 3:2 ratio

The evolution of multiple stellar systems can be driven by Kozai cycles and tidal friction (KCTF), which shrink the orbit of the inner binary. There is an interesting possibility that two close binaries on a common long-period orbit experience mutually-induced KCTF. We present the discovery of a possible new quadruple system composed of two unresolved eclipsing binaries (EBs), CzeV343 (V~13.5 mag). We obtained photometric observations of CzeV343 that completely cover the two orbital periods and we successfully model the light curves as the sum of two detached EBs. We provide confidence intervals for the model parameters and minima timings by bootstrap resampling of our data. One of the EBs shows a distinctly eccentric orbit with a total eccentricity of about 0.18. The two orbital periods, 1.20937 and 0.80693 days, are within 0.1% of a 3:2 ratio. We speculate that this might be the result of KCTF-driven evolution of a quadruple system and we discuss this hypothesis in the context of other quadruple systems composed of two EBs. We make our double EB fitting code publicly available to provide a tool for long-term monitoring of the mutual orbit in such systems.

Physical identification of static and axially-symmetric vacuum metric tensors

We present a method of a possible physical identification of the static and axially symmetric Weyl-type vacuum γ and nγ metrics. This method, in which no interior solutions of any kind are involved, is based on the comparison of the far-field forms of the γ and nγ metrics, and of the far-field form of the metric tensor due to a bounded gravitating perfect-fluid source given correctly to post-Newtonian accuracy. The parameters of the vacuum solutions are expressed in terms of physical parameters of a prolate fluid source, namely total mass-energy, semiaxes and eccentricity, defined consistently to post-Newtonian accuracy. The results, based on the otherwise arbitrary fluid source, appear physically general. Possible astrophysical candidates for the far-field γ andnγ metrics are proposed, based on the conditions imposed on them by the identification method. The advantages and deficiencies of the identification method are briefly discussed.

On the Behavior of Gas-Lubricated Journal Bearings Subjected to Sinusoidally Time-Varying Loads

Linearized steady-state time-dependent solutions to the equations of journal motion are obtained for sinusoidally time-varying radial loads by utilizing the “linearized ph” technique to approximate lubricant pressure forces. An exemplary check for vibration at half-rotor frequency shows that the nonlinear terms can be neglected provided the total eccentricity ratio (static plus dynamic) remains less than one half. A typical journal-bearing frequency response exhibits two distinct types of resonances: One at half-rotor frequency and another at a frequency given by K/M, where K is the effective spring constant due to the bulk modulus of the gas lubricant and M is the mass of the supported rotor. As the static eccentricity ratio ∈0 increases, the amplitude of the half-rotor frequency resonance decreases drastically, but the amplitude of the K/M resonance increases slightly. Rotating load response can be synthesized by superimposing the responses to two 90 deg out-of-phase radial loads acting along perpendicular axes. The resulting response is a nearly circular ellipse centered about the static equilibrium position.