Lower bounds of the Laplacian spectrum of graphs based on diameter

Let G be a connected graph of order n. The diameter of G is the maximum distance between any two vertices of G. In the paper, we will give some lower bounds for the algebraic connectivity and the Laplacian spectral radius of G in terms of the diameter of G.     

On the characterization of the domination of a diameter-constrained network reliability model

Let G=(V,E) be a digraph with a distinguished set of terminal vertices K⊆V and a vertex s∈K. We define the s,K-diameter of G as the maximum distance between s and any of the vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the diameter-constrained s,K-terminal reliability of G, R_s,K(G,D), is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D.The diameter-constrained network reliability is a special case of coherent system models, where the domination invariant has played an important role, both theoretically and for developing algorithms for reliability computation. In this work, we completely characterize the domination of diameter-constrained network models, giving a simple rule for computing its value: if the digraph either has an irrelevant arc, includes a directed cycle or includes a dipath from s to a node in K longer than D, its domination is 0; otherwise, its domination is -1 to the power |E|-|V|+1. In particular this characterization yields the classical source-to-K-terminal reliability domination obtained by Satyanarayana.Based on these theoretical results, we present an algorithm for computing the reliability.     

Parameter study and characterization for polyacrylonitrile nanofibers fabricated via centrifugal spinning process

Electrospinning is currently the most popular method for producing polymer nanofibers. However, the low production rate and safety concern limit the practical use of electrospinning as a cost-effective nanofiber fabrication approach. Herein, we present a novel and simple centrifugal spinning technology that extrudes nanofibers from polymer solutions by using a high-speed rotary and perforated spinneret. Polyacrylonitrile (PAN) nanofibers were prepared by selectively varying parameters that can affect solution intrinsic properties and operational conditions. The resultant PAN nanofibers were characterized by SEM, and XRD. The correlation between fiber morphology and processing conditions was established. Results demonstrated that the fiber morphology can be easily manipulated by controlling the spinning parameters and the centrifugal spinning process is a facile approach for fabricating polymer nanofibers in a large-scale and low-cost fashion.     

On the eccentric distance sum of trees and unicyclic graphs

Let G be a simple connected graph with the vertex set V(G). The eccentric distance sum of G is defined as ξd(G)=_∑v∈V(G)ε(v)DG(v), where ε(v) is the eccentricity of the vertex v and DG(v)=_∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum.     

On the distance distribution of trees

Let G be a tree and k a non-negative integer. We determine best possible upper and lower bounds on the number of pairs of vertices at distance exactly k in G in terms of order alone, and in terms of order and radius or diameter.     

A cooperative location game based on the 1-center location problem

In this paper we introduce and analyze new classes of cooperative games related to facility location models defined on general metric spaces. The players are the customers (demand points) in the location problem and the characteristic value of a coalition is the cost of serving its members. Specifically, the cost in our games is the service radius of the coalition. We call these games the Minimum Radius Location Games (MRLG).We study the existence of core allocations and the existence of polynomial representations of the cores of these games, focusing on network spaces, i.e., finite metric spaces induced by undirected graphs and positive edge lengths, and on the ?_p metric spaces defined over R^d.     

Reduced convex bodies in Euclidean space—A survey

A convex body R in Euclidean space E^d is called reduced if the minimal width Δ(K) of each convex body K⊂R different from R is smaller than Δ(R). This definition yields a class of convex bodies which contains the class of complete sets, i.e., the family of bodies of constant width. Other obvious examples in E² are regular odd-gons. We know a relatively large amount on reduced convex bodies in E². Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For d≥3 we do not even have tools which permit us to recognize what the boundary of R looks like. The class of reduced convex bodies has interesting applications. We present the current state of knowledge about reduced convex bodies in E^d, recall some striking related research problems, and put a few new questions.     

Lengths of Paths and Cycles in Zero-Divisor Graphs and Digraphs of Semigroups

Short, elementary proofs of the optimal bounds for various path and cycle lengths are given for the zero-divisor graphs and digraphs of semigroups.