a new family of extended gauss quadratures with an interior interval constraint starting from two sequences ga c n and gd b n of ordinary gauss quadrature formulae with an orthogonality measure d on the open intervals a c and d b respectively we construct a new sequence ga b e n of extended gaussian quadrature formulae for d on a b which is based on some preassigned points the nodes of ga c n gd b n and the e n zeros contained in c d of a nonclassical orthogonal polynomial on a b with respect to a linear functional the principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three term recurrence formulae are related thus a new class of gaussian quadratures having some nodes contained in a given interior interval can be computed directly by standard software for ordinary gauss quadrature formulae

odd cycles embedding on folded hypercubes with conditional faulty edges let ff e be the set of ff e 4 and n is even under this assumption we show that every fault free edge of fq n lies on a fault free cycle of every odd length from n 1 to 2 n 1 in terms of the number of tolerant faulty edges and embedding odd cycles in fq n our result improves not only the result in xu and ma 2006 where ff e 0 but also the previous best result gotten by xu et al 2006 where ff e

many to many two disjoint path covers in restricted hypercube like graphs a disjoint path cover dpc for short of a graph is a set of pairwise internally disjoint paths that altogether cover every vertex of the graph given a set s of k sources and a set t of k sinks a many to many k dpc between s and t is a disjoint path cover each of whose paths joins a pair of source and sink it is classied as paired if each source of s must be joined to a designated sink oft or unpaired if there is no such constraint in this paper we show that every m dimensional restricted hypercube like graph with at most m 3 faulty vertices and or edges being removed has a paired and unpaired 2 dpc joining arbitrary two sources and two sinks where m 5 the bound m 3 on the number of faults is optimal for both paired and unpaired types

paired many to many disjoint path covers in faulty hypercubes a paired many to many k disjoint path cover k dpc for short of a graph is a set of k disjoint paths joining k distinct source sink pairs that cover all the vertices of the graph extending the notion of dpc we define a paired many to many bipartite k dpc of a bipartite graph g to be a set of k disjoint paths joining k distinct source sink pairs that altogether cover the same number of vertices as the maximum number of vertices covered when the source sink pairs are given in the complete bipartite spanning supergraph of g we show that every m dimensional hypercube q m under the condition that f or less faulty elements vertices and or edges are removed has a paired many to many bipartite k dpc joining any k distinct source sink pairs for any f and k 1 subject to f 2k

paired many to many disjoint path covers of the hypercubes in this paper we consider the problem of paired many to many disjoint path covers of the hypercubes and obtain the following result let s s 1 s 2 s k and t t 1 t 2 t k be two sets of k vertices in different partite sets of the n dimensional hypercube q n and let e i s iandt iare adjacent 1 email protected k e email protected then there exist k vertex disjoint paths p 1 p 2 p k where p i connects s i and t i for i 1 2 k such that these k paths contain all vertices of q n

on mean convergence of extended lagrange interpolation abstract lagrange interpolation to any continuous function on 1 1 at the zeros of orthogonal polynomials is known to converge in the mean here following bellen we study mean convergence of lagrange interpolation on an extended set of nodes that includes in addition to the n zeros of the orthagonal relative to some positive weight function w polynomial n of degree n other n 1 nodes which in turn are zeros of an orthogonal polynomial gp n 1 of degree n 1 corresponding to the weight function w n 2 n w a sufficient criterion of bellen for mean convergence as n of such extended lagrange interpolation for arbitrary continuous functions is shown to fail for chebyshev weight functions of the first third and fourth kind it holds trivially for chebyshev weights of the second kind based on extensive computations it is conjectured on the other hand that the criterion is satisfied for certain jacobi weights with parameters and suitably restricted necessary conditions for mean convergence due to erd s and turan are shown to be violated for the three kinds of chebyshev weights mentioned above for smooth functions a comparison is made of the speed of convergence of simple vs extended lagrange interpolation

on the spanning connectivity and spanning laceability of hypercube like networks let u and v be any two distinct nodes of an undirected graph g which is k connected for email protected email protected k a w containerc u v of a k connected graph g is a set of w disjoint paths joining u and v a w container c u v of g is a w container if it contains all the nodes of g a graph g is w connected if there exists a w container between any two distinct nodes a bipartite graph g is w laceable if there exists a w container between any two nodes from different parts of g let g 0 v 0 e 0 and g 1 v 1 e 1 be two disjoint graphs with v 0 v 1 let e v f v email protected v 0 f v v 1 and f v 0 v 1 is a bijection let g g email protected g 1 v email protected v 1 e email protected e email protected e the set of n dimensional hypercube like graph h n is defined recursively as a h 1 k 2 k 2 complete graph with two nodes and b if g 0 and g 1 are in h n then g g email protected g 1 is in h n 1 let b n email protected h n and g is bipartite and n n h n b n in this paper we show that every graph in b n is w laceable for every w email protected email protected n it is shown that a constructed n n graph h can not be 4 connected in addition we show that every graph in n n is w connected for every w email protected email protected 3

on quadrature convergence of extended lagrange interpolation quadrature convergence of the extended lagrange interpolant l 2n 1 f for any continuous function f is studied where the interpolation nodes are the n zeros i of an orthogonal polynomial of degree n and the n 1 zeros j of the corresponding induced orthogonal polynomial of degree n 1 it is found that unlike convergence in the mean quadrature convergence does hold for all four chebyshev weight functions this is shown by establishing the positivity of the underlying quadrature rule whose weights are obtained explicitly necessary and sufficient conditions for positivity are also obtained in cases where the nodes i and j interlace and the conditions are checked numerically for the jacobi weight function with parameters and it is conjectured in this case that quadrature convergence holds for 1 2 1 2

two conditions for reducing the maximal length of node disjoint paths in hypercubes efficient methods have been developed for constructing m node disjoint paths from one source node to other m not necessarily distinct destination nodes in an n dimensional hypercube so that not only is their total length minimized but their maximal length is also minimized in the worst case where m n for general case their maximal length is not greater than the minimum of n 1 and the maximal distance between the source node and destination nodes plus two in this paper we show that their maximal length can be further reduced by at least 1 if one of the two conditions holds besides their total length remains minimum and each path remains either shortest or second shortest in the situation that all of the source node and destination nodes are mutually distinct computer simulation results show that by excluding two trivial worst cases in which the maximal length of previously constructed node disjoint paths from the source node to the destination nodes is not only greater than the maximal distance but also impossible to be further reduced the probability that one of the two conditions holds is greater than 71 73 79 and 85 for m n 4 5 6 and 7 respectively

many to many disjoint path covers in the presence of faulty elements a many to many k disjoint path cover k dpc of a graph g is a set of k disjoint paths joining k sources and k sinks in which each vertex of g is covered by a path it is called a paired many to many disjoint path cover when each source should be joined to a specific sink and it is called an unpaired many to many disjoint path cover when each source can be joined to an arbitrary sink in this paper we discuss about paired and unpaired many to many disjoint path covers including their relationships application to strong hamiltonicity and necessary conditions and then we give a construction scheme for paired many to many disjoint path covers in the graph h0 oplus h1 obtained from connecting two graphs h0 and h1 with v h0 v h1 by v h1 pairwise nonadjacent edges joining vertices in h0 and vertices in h1 where h0 g0 oplus g1 and h1 g2 oplus g3 for some graphs gj using the construction we show that every m dimensional restricted hl graph and recursive circulant g 2m 4 with f or less faulty elements have a paired k dpc for any f and k ges 2 with f 2k les m

local neighbourhoods for first passage percolation on the configuration model we consider first passage percolation on the configuration model once the network has been generated each edge is assigned an i i d weight modeling the passage time of a message along this edge then independently two vertices are chosen uniformly at random a sender and a recipient and all edges along the geodesic connecting the two vertices are coloured in red in the case that both vertices are in the same component in this article we prove local limit theorems for the coloured graph around the recipient in the spirit of benjamini and schramm we consider the explosive regime in which case the random distances are of finite order and the malthusian regime in which case the random distances are of logarithmic order

long paths and cycles in hypercubes with faulty vertices a fault free path in the n dimensional hypercube q n with f faulty vertices is said to be long if it has length at least 2 n 2f 2 similarly a fault free cycle in q n is long if it has length at least 2 n 2f if all faulty vertices are from the same bipartite class of q n such length is the best possible we show that for every set of at most 2n 4 faulty vertices in q n and every two fault free vertices u and v satisfying a simple necessary condition on neighbors of u and v there exists a long fault free path between u and v this number of faulty vertices is tight and improves the previously known results furthermore we show for every set of at most n 2 10 n 2 1 faulty vertices in q n where n 15 that q n has a long fault free cycle this is a first quadratic bound which is known to be asymptotically optimal

hamiltonian paths and cycles with prescribed edges in the 3 ary n cube the k ary n cube has been one of the most popular interconnection networks for massively parallel systems given a set p of at most 2n 2 n 2 prescribed edges and two vertices u and v we show that the 3 ary n cube contains a hamiltonian path between u and v passing through all edges of p if and only if the subgraph induced by p consists of pairwise vertex disjoint paths none of them having u or v as internal vertices or both of them as end vertices as an immediate result the 3 ary n cube contains a hamiltonian cycle passing through a set p of at most 2n 1 prescribed edges if and only if the subgraph induced by p consists of pairwise vertex disjoint paths

hamiltonian path embeddings in conditional faulty k ary n cubes the class of k ary n cubes represents the most commonly used interconnection topology for distributed memory parallel systems a k ary n cube is bipartite if and only if k is even in this paper we consider the faulty k ary n cube with even k 4 and n 2 such that each vertex of the k ary n cube is incident with at least two healthy edges based on this requirement we prove that the k ary n cube contains a hamiltonian path joining every pair of vertices which are in different parts even if it has up to 4n 5 edge faults and this result is optimal

long paths in hypercubes with conditional node faults let f be a set of f 2 n 5 faulty nodes in an n cube q n such that every node of q n still has at least two fault free neighbors then we show that q n f contains a path of length at least 2 n 2 f 1 respectively 2 n 2 f 2 between any two nodes of odd respectively even distance since the n cube is bipartite the path of length 2 n 2 f 1 or 2 n 2 f 2 turns out to be the longest if all faulty nodes belong to the same partite set as a contribution our study improves upon the previous result presented by j s fu longest fault free paths in hypercubes with vertex faults information sciences 176 2006 759 771 where only n 2 faulty nodes are considered

paired 2 disjoint path covers of faulty k ary n cubes a paired k disjoint path cover of a graph is a set of k disjoint paths joining k distinct source sink pairs that cover all vertices of the graph the k ary n cube q n k is one of the most popular interconnection networks in this paper we consider the problem of paired 2 disjoint path covers of the k ary n cube q n k odd k 5 with faulty elements vertices and or edges and obtain the following result let f be a set of faulty elements in q n k odd k 5 with f 2 n 4 and a b and c d be its any two pairs of non faulty vertices then the graph q n k f contains vertex disjoint a b path and c d path that cover its all non faulty vertices and the upper bound 2 n 4 of faults tolerated is nearly optimal we also show that cartesian product q n k p with at most 2 n 2 faulty elements is hamiltonian connected where p is a path n 2 and odd k 5 a paired k dpc of a graph is a set of k disjoint paths joining k source sink pairs that cover all vertices of the graph let q n k be the k ary n cube and f be a set of faulty elements where odd k 5 and f 2 n 4 let a b and c d be any two pairs of non faulty vertices then q n k f has a 2 dpc a b path and c d path the upper bound 2 n 4 of faults tolerated is nearly optimal

first passage percolation on locally treelike networks i dense random graphs we study various properties of least cost paths under independent and identically distributed iid disorder for the complete graph and dense erd s renyi random graphs in the connected phase with iid exponential and uniform weights on edges using a simple heuristic we compute explicitly limiting distributions for properly recentered lengths of shortest paths between typical nodes as well as multiple source destination pairs we also derive asymptotics for the number of edges on the shortest path namely the hopcount and find that the addition of edge weights converts these graphs from ultrasmall world networks to small world networks finally we study the vickrey clarke groves measure of overpayment for the complete graph with exponential edge weights and show that the complete graph is far from monopolistic for large n

constructing vertex disjoint paths in n k star graphs this work describes a novel routing algorithm for constructing a container of width n 1 between a pair of vertices in an n k star graph with connectivity n 1 since lin et al t c lin d r duh h c cheng wide diameter of n k star networks in proceedings of the international conference on computing communications and control technologies vol 5 2004 pp 160 165 already calculated the wide diameters in n n 1 star and n 1 star graphs this study only considers an n k star with 2

many to many disjoint paths in faulty hypercubes this paper considers the problem of many to many disjoint paths in the hypercube q n with f v faulty vertices and f e faulty edges and obtains the following result for any integer k with 1 k n 1 any two sets s and t of k fault free vertices in different parts if f v f e n k 1 then there exist k disjoint fault free s t paths in q n which contains at least 2 n 2 f v vertices this result is optimal in the worst case

unpaired many to many disjoint path covers in restricted hypercube like graphs for two disjoint vertex sets s s 1 s k and t t 1 t k of a graph an unpaired many to many k disjoint path cover joining s and t is a set of pairwise vertex disjoint paths p 1 p k that altogether cover every vertex of the graph in which p i is a path from s i to some t j for 1 i j k a family of hypercube like interconnection networks called restricted hypercube like graphs includes most nonbipartite hypercube like networks found in the literature such as twisted cubes crossed cubes mobius cubes recursive circulant g 2 m 4 of odd m etc in this paper we show that every m dimensional restricted hypercube like graph m 5 with at most f faulty vertices and or edges being removed has an unpaired many to many k disjoint path cover joining arbitrary disjoint sets s and t of size k each subject to k 2 and f k m 1 the bound m 1 on f k is the maximum possible

a faber series approach to cardinal interpolation for a compactly supported function 9 in rd we study quasiinterpolants based on point evaluations at the integer lattice we restrict ourselves to the case where the coefficient sequence af for given data f is computed by applying a univariate polynomial q to the sequence i zd and then convolving with the data flzd such operators appear in the well known neumann series formulation of quasi interpolation a criterion for the polynomial q is given such that the corresponding operator defines a quasi interpolant since our main application is cardinal interpolation which is well defined if the symbol of p does not vanish we choose q as the partial sum of a certain faber series this series can be computed recursively by this approach we avoid the restriction that the range of the symbol of p must be contained in a disk of the complex plane excluding the origin which is necessary for convergence of the neumann series furthermore for symmetric a we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the neumann series

paired 2 disjoint path covers and strongly hamiltonian laceability of bipartite hypercube like graphs abstract a paired many to many k disjoint path cover paired k dpc for short of a graph is a set of k vertex disjoint paths joining k distinct source sink pairs that altogether cover every vertex of the graph we consider the problem of constructing paired 2 dpc s in an m dimensional bipartite hl graph x m and its application in finding the longest possible paths it is proved that every x m m 4 has a fault free paired 2 dpc if there are at most m 3 faulty edges and the set of sources and sinks is balanced in the sense that it contains the same number of vertices from each part of the bipartition furthermore every x m m 4 has a paired 2 dpc in which the two paths have the same length if each source sink pair is balanced using 2 dpc properties we show that every x m m 3 with either at most m 2 faulty edges or one faulty vertex and at most m 3 faulty edges is strongly hamiltonian laceable

paired many to many disjoint path covers in recursive circulants g 2 m 4 a disjoint path cover dpc for short of a graph is a set of disjoint paths that cover all the vertices of the graph a paired many to many k dpc is a dpc composed of k paths between k sources and k sinks such that each source is joined to a designated sink we show that recursive circulant g 2m 4 with at most f faulty vertices and or edges being removed has a paired many to many k dpc joining k arbitrary sources and sinks for any f and k 2 subject to f 2k m 1 where m 5 the bound m 1 on f 2k is the best possible