The local spectra of line graphs

The local spectrum of a graph G = (V, E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is “seen” from a given vertex. Thus, for each vertex i ∈ V , the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λl ∈ ev G is the sum, extended to all vertices, of its local multiplicities. In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LG in terms of the local spectrum of the (regular o semiregular) graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results related to the number of cycles

Sampling cluster endurance for peer-to-peer based content distribution networks

Several types of Content Distribution Networks are being deployed over the Internet today, based on different architectures to meet their requirements (e.g., scalability, efficiency and resiliency). Peer-to-peer (P2P) based Content Distribution Networks are promising approaches that have several advantages. Structured P2P networks, for instance, take a proactive approach and provide efficient routing mechanisms. Nevertheless, their maintenance can increase considerably in highly dynamic P2P environments. In order to address this issue, a two-tier architecture called Omicron that combines a structured overlay network with a clustering mechanism is suggested in a hybrid scheme. In this paper, we examine several sampling algorithms utilized in the aforementioned hybrid network that collect local information in order to apply a selective join procedure. Additionally, we apply the sampling algorithms on Chord in order to evaluate sampling as a general information gathering mechanism. The algorithms are based mostly on random walks inside the overlay networks. The aim of the selective join procedure is to provide a well balanced and stable overlay infrastructure that can easily overcome the unreliable behavior of the autonomous peers that constitute the network. The sampling algorithms are evaluated using simulation experiments as well as probabilistic analysis where several properties related to the graph structure are reveale

On cyclic Kautz digraphs

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d, `) and it is derived from the Kautz digraphs K(d, `). It is well-known that the Kautz digraphs K(d, `) have the smallest diameter among all digraphs with their number of vertices and degree. We define the cyclic Kautz digraphs CK(d, `), whose vertices are labeled by all possible sequences a1 . . . a` of length `, such that each character ai is chosen from an alphabet containing d + 1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a1 6= a`. The cyclic Kautz digraphs CK(d, `) have arcs between vertices a1a2 . . . a` and a2 . . . a`a`+1, with a1 6= a` and a2 6= a`+1. Unlike in Kautz digraphs K(d, `), any label of a vertex of CK(d, `) can be cyclically shifted to form again a label of a vertex of CK(d, `). We give the main parameters of CK(d, `): number of vertices, number of arcs, and diameter. Moreover, we construct the modified cyclic Kautz digraphs MCK(d, `) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d, `) are d-out-regular. Finally, we compute the number of vertices of the iterated line digraphs of CK(d, `).Preprin

Redes de interconexión: contribución al estudio de su vulnerabilidad

Esta tesis se ha centrado en el estudio de la vulnerabilidad del encaminamiento en familias de grafos y dígrafos que resultan particularmente interesantes por su aplicación al diseño de redes de interconexión: redes asociadas con teselaciones del plano, dígrafos línea parciales de Kautz y de Brujin. Se demuestra la existencia de vértices (p,f)-centrales en los grafos de doble y triple lazo para determinados f y se determinan conjuntos de vértices y conjuntos de ramas que pueden fallar sin que se pierda la comunicación entre los restantes en las dos familias anteriores y en los dígrafos bipartitos bd (s,b,c,d). Este estudio se hace a partir de la representación geométrica que representan estas redes asociadas a teselaciones del plano.También se da un método que construye toda la sucesión finita de dígrafos línea parciales de Kautz y de Brujin presentando un algoritmo que calcula vértices (p,f)-centrales en estas redes

The partial line digraph technique in the design of large interconnection networks

The following problem arises in the design of some interconnection networks for distributed systems. Namely, to construct digraphs with given maximum out-degree, reduced diameter, easy routing, good connectivity, and good expandability. To this end, a new method based on the concept of partial line digraph is presented. This proposal, which turns out to be a generalization of the so-called line digraph technique, allows us to obtain digraphs that satisfy all the above-mentioned requirements. In particular it is shown that the partial line digraphs of Kautz digraphs solve the (d. digraph problem-i.e. to minimize the diameter D in a digraph of maximum outdegree d and number of vertices S-for any in the range Index Terms- Communication network, connectivity maximization, (d. Y) digraph problem, expandability, fault-tolerance, line digraph, routing algorithm. an-l + (P2+ . . . + 1 5 -1- 5 drl + P1.Postprint (published version

The partial line digraph technique in the design of large interconnection networks

The following problem arises in the design of some interconnection networks for distributed systems. Namely, to construct digraphs with given maximum out-degree, reduced diameter, easy routing, good connectivity, and good expandability. To this end, a new method based on the concept of partial line digraph is presented. This proposal, which turns out to be a generalization of the so-called line digraph technique, allows us to obtain digraphs that satisfy all the above-mentioned requirements. In particular it is shown that the partial line digraphs of Kautz digraphs solve the (d. digraph problem-i.e. to minimize the diameter D in a digraph of maximum outdegree d and number of vertices S-for any in the range Index Terms- Communication network, connectivity maximization, (d. Y) digraph problem, expandability, fault-tolerance, line digraph, routing algorithm. an-l + (P2+ . . . + 1 5 -1- 5 drl + P1

Structural properties and labeling of graphs

The complexity in building massive scale parallel processing systems has re- sulted in a growing interest in the study of interconnection networks design. Network design affects the performance, cost, scalability, and availability of parallel computers. Therefore, discovering a good structure of the network is one of the basic issues. From modeling point of view, the structure of networks can be naturally stud- ied in terms of graph theory. Several common desirable features of networks, such as large number of processing elements, good throughput, short data com- munication delay, modularity, good fault tolerance and diameter vulnerability correspond to properties of the underlying graphs of networks, including large number of vertices, small diameter, high connectivity and overall balance (or regularity) of the graph or digraph. The first part of this thesis deals with the issue of interconnection networks ad- dressing system. From graph theory point of view, this issue is mainly related to a graph labeling. We investigate a special family of graph labeling, namely antimagic labeling of a class of disconnected graphs. We present new results in super (a; d)-edge antimagic total labeling for disjoint union of multiple copies of special families of graphs. The second part of this thesis deals with the issue of regularity of digraphs with the number of vertices close to the upper bound, called the Moore bound, which is unobtainable for most values of out-degree and diameter. Regularity of the underlying graph of a network is often considered to be essential since the flow of messages and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element. This means that the in-degree and out-degree of each processing element must be the same or almost the same. Our new results show that digraphs of order two less than Moore bound are either diregular or almost diregular.Doctor of Philosoph

Fuzzy EOQ Model with Trapezoidal and Triangular Functions Using Partial Backorder

EOQ fuzzy model is EOQ model that can estimate the cost from existing information. Using trapezoid fuzzy functions can estimate the costs of existing and trapezoid membership functions has some points that have a value of membership . TR ̃C value results of trapezoid fuzzy will be higher than usual TRC value results of EOQ model . This paper aims to determine the optimal amount of inventory in the company, namely optimal Q and optimal V, using the model of partial backorder will be known optimal Q and V for the optimal number of units each time a message . EOQ model effect on inventory very closely by using EOQ fuzzy model with triangular and trapezoid membership functions with partial backorder. Optimal Q and optimal V values for the optimal fuzzy models will have an increase due to the use of trapezoid and triangular membership functions that have a different value depending on the requirements of each membership function value. Therefore, by using a fuzzy model can solve the company's problems in estimating the costs for the next term