the spectral radius of tricyclic graphs with n vertices and k pendent vertices in this paper we determine graphs with the largest spectral radius among all the tricyclic graphs with n vertices and k pendent vertices

on the spectral radius of trees with fixed diameter let t n d be the set of trees on n vertices with diameter d in this paper the first d2 1 spectral radii of trees in the set t n d 3 d n 4 are characterized

nonequiprobable signaling on the gaussian channel signaling schemes for the gaussian channel based on finite dimensional lattices are considered the signal constellation consists of all lattice points within a region r and the shape of this region determines the average signal power spherical signal constellations minimize average signal power and in the limit as n to infinity the shape gain of the n sphere over the n cube approaches pi e 6 approximately 1 53 db a nonequiprobable signaling scheme is described that approaches this full asymptotic shape gain in any fixed dimension a signal constellation omega is partitioned into t subconstellations omega sub 0 omega sub tau 1 of equal size by scaling a basic region r signal points in the same subconstellation are used equiprobably and a shaping code selects the subconstellation omega sub i with frequency f sub i shaping codes make it possible to achieve any desired fractional bit rate the schemes presented are compared with equiprobable signaling schemes based on voronoi regions of multidimensional lattices for comparable shape gain and constellation expansion ratio the peak to average power ratio of the schemes presented is superior furthermore a simple table lookup is all that is required to address points in the constellations it is also shown that it is possible to integrate coding and nonequiprobable signaling within a common multilevel framework

on the spectral radius of trees with given independence number abstract in the paper we will determine the graphs with maximal spectral radius among all the trees with n vertices and independence number for n 2 n 1

on the spectral radius of bipartite graphs with given diameter let gb n d be the set of bipartite graphs with order n and diameter d this paper characterizes the extremal graph with the maximal spectral radius in gb n d furthermore the maximal spectral radius is a decreasing function on d at last bipartite graphs with the second largest spectral radius are determined

shaping multidimensional signal spaces i optimum shaping shell mapping the structure of the regions which provide the optimum tradeoff between gamma sub s shaping gain and cer constellation expansion ratio and between gamma sub s and par peak to average power ratio in a finite dimensional space is introduced analytical expressions are derived for the corresponding tradeoff curves in general the initial parts of the curves have a steep slope this means that an appreciable portion of the maximum shaping gain corresponding to a spherical region can be achieved with a small value of cer sub s par the technique of shell mapping is introduced this is a change of variable which maps the optimum shaping region to a hypercube truncated within a simplex this mapping is a useful tool in computing the performance and also in facilitating the addressing of the optimum shaping region using the shell mapping a practical addressing scheme is presented that achieves a point on the optimum tradeoff curves for dimensionalities around 12 the point achieved is located near the knee of the corresponding tradeoff curve for larger dimensionalities a general shaping region with two degrees of freedom is used this region provides more flexibility in selecting the tradeoff point

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

shaping of multidimensional signal constellations using a lookup table this paper describes a lookup table for the addressing of an optimally shaped constellation the method is based on partitioning the subconstellations into shaping macro shells of integer bit rate and increasing average energy the macro shells do not need to have an equal number of points a lookup table is used to select a subset of the partitions in the cartesian product space by devising appropriate partitioning merging rules we obtain suboptimum schemes of very low addressing complexity and small performance degradation the performance is computed using the weight distribution of an optimally shaped constellation

tables of sphere packings and spherical codes the theta function of a sphere packing gives the number of centers at each distance from the origin the theta functions of a number of important packings a n d n e n the leech lattice and others and tables of the first fifty or so of their coefficients are given in this paper

new upper bounds on the spectral radius of trees with the given number of vertices and maximum degree abstract this paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree we obtain the new upper bounds on the spectral radius of the trees and the results are the best upper bounds expressed by the number of vertices and maximum degree at present let t v e be a tree on n vertices with maximum degree where 3 n 2 denote by t the spectral radius of t we prove that 1 if n 2 then t n 1 n 2 2 2 n 3 2 and equality holds if and only if t is an almost completely full degree tree of 3 levels 2 if 2 n 2 1 then t 2 1 and equality holds if and only if t is a completely full degree tree of 3 levels 3 if n 2 1 then t 2 1 cos 2 k 1 where k log 1 2 n 1 1 1

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

new rotationally invariant four dimensional trellis codes two new classes of rotationally invariant trellis codes are constructed a simple method is used to check the rotational invariance of a given code in the process of searching for optimum trellis codes a class of linear trellis codes with constraint lengths 2 9 using four dimensional constellations is presented simulation results show that a 180 spl deg rotationally invariant constraint length 8 linear trellis code achieves about 0 4 db real coding gain compared to the best constraint length 6 code while the trellis complexity is only four times that of the constraint length 6 code on the other hand the constraint length 6 code that has been adopted for use in the v 34 28 8 kbit s modem standard has the same 0 4 db real coding gain compared to a constraint length 4 code which has also been adopted for the standard but it requires sixteen times the trellis complexity a class of fully rotationally invariant nonlinear trellis codes with constraint lengths 6 11 is also presented simulation results show that a 90 spl deg rotationally invariant constraint length 8 nonlinear trellis code performs almost as well as the best linear code

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

the perron eigenspace of nonnegative almost skew symmetric matrices and levinger s transformation let a be a nonnegative square matrix whose symmetric part has rank one tournament matrices are of this type up to a positive shift by 1 2i when the symmetric part of a is irreducible the perron value and the left and right perron vectors of l a 1 a at are studied and compared as functions of 0 1 2 in particular upper bounds are obtained for both the perron value and its derivative as functions of the parameter via the notion of the q numerical range

the first two largest spectral radii of uniform supertrees with given diameter abstract a supertree is a connected and acyclic hypergraph let s m d k be the set of k uniform supertrees with m edges and diameter d in this paper we determine the supertrees with the first two largest spectral radii among all supertrees in s m d k for 3 d m 1

on optimal shaping of multidimensional constellations a scheme for the optimal shaping of multidimensional constellations is proposed this scheme is motivated by a type of structured vector quantizer for memoryless sources and results in n sphere shaping of n dimensional cubic lattice based constellations because n sphere shaping is optimal in n dimensions shaping gains higher than those of n dimensional voronoi constellations can be realized while optimal shaping for a large n can realize most of the 1 53 db total shaping gain it has the undesirable effect of increasing the size and the peak to average power ratio of the constituent 2d constellation this limits its usefulness for many real world channels which have nonlinearities the proposed scheme alleviates this problem by achieving optimal constellation shapes for a given limit on the constellation expansion ratio or the peak to average power ratio of the constituent 2d constellation results of calderbank and ozarow 1990 on nonequiprobable signaling are used to reduce the complexity of this scheme and make it independent of the data rate with essentially no effect on the shaping gain comparisons with forney s 1989 trellis shaping scheme are also provided

optimal nonuniform signaling for gaussian channels variable rate data transmission schemes in which constellation points are selected according to a nonuniform probability distribution are studied when the criterion is one of minimizing the average transmitted energy for a given average bit rate the best possible distribution with which to select constellations points is a maxwell boltzmann distribution in principle when constellation points are selected according to a maxwell boltzmann distribution the ultimate shaping gain pi e 6 or 1 53 db can be achieved in any dimension nonuniform signaling schemes can be designed by mapping simple variable length prefix codes onto the constellation using the huffman procedure prefix codes can be designed that approach the optimal performance these schemes provide a fixed rate primary channel and a variable rate secondary channel and are easily incorporated into standard lattice type coded modulation schemes

on spectral radius of graphs with pendant paths abstract the spectral radius of a graph is the largest eigenvalue of the adjacency matrix corresponding to this graph let g be a graph with long pendant paths in this note we estimate the spectral radius of g by obtaining appropriate upper and lower bounds as an application the spectral radius of some k ary trees are estimated

the minimal spectral radius of graphs with a given diameter the spectral radius of a graph i e the largest eigenvalue of its corresponding adjacency matrix plays an important role in modeling virus propagation in networks in fact the smaller the spectral radius the larger the robustness of a network against the spread of viruses among all connected graphs on n nodes the path pn has minimal spectral radius however its diameter d i e the maximum number of hops between any pair of nodes in the graph is the largest possible namely d n 1 in general communication networks are designed such that the diameter is small because the larger the number of nodes traversed on a connection the lower the quality of the service running over the network this leads us to state the following problem which connected graph on n nodes and a given diameter d has minimal spectral radius in this paper we solve this problem explicitly for graphs with diameter d fenced 1 2 fenced frac n 2 n 3 n 2 n 1 moreover we solve the problem for almost all graphs on at most 20 nodes by a computer search 2007 elsevier inc all rights reserved

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

optical intensity modulated direct detection channels signal space and lattice codes traditional approaches to constructing constellations for electrical channels cannot be applied directly to the optical intensity channel this work presents a structured signal space model for optical intensity channels where the nonnegativity and average amplitude constraints are represented geometrically lattice codes satisfying channel constraints are defined and coding and shaping gain relative to a baseline are computed an effective signal space dimension is defined to represent the precise impact of coding and shaping on bandwidth average optical power minimizing shaping regions are derived in some special cases example lattice codes are constructed and their performance on an idealized point to point wireless optical link is computed bandwidth efficient schemes are shown to have promise for high data rate applications but require greater average optical power

the minimal laplacian spectral radius of trees with a given diameter for a graph g its laplacian matrix is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix let t n d be the set of trees on n vertices with diameter d in this paper for email protected 1 2 3 4 n 3 n 2 n 1 trees with minimal laplacian spectral radii in the set t n d are characterized

the least eigenvalue of graphs with given connectivity abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of g are those of a g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity

near capacity coding in multicarrier modulation systems we apply irregular low density parity check ldpc codes to the design of multilevel coded quadrature amplitude modulation qam schemes for application in discrete multitone systems in frequency selective channels a combined gray ungerboeck scheme is used to label each qam constellation the gray labeled bits are protected using an irregular ldpc code with iterative soft decision decoding while other bits are protected using a high rate reed solomon code with hard decision decoding or are left uncoded the rate of the ldpc code is selected by analyzing the capacity of the channel seen by the gray labeled bits and is made adaptive by selective concatenation with an inner repetition code using a practical bit loading algorithm we apply this coding scheme to an ensemble of frequency selective channels with gaussian noise over a large number of channel realizations this coding scheme provides an average effective coding gain of more than 7 5 db at a bit error rate of 10 sup 7 and a block length of approximately 10 sup 5 b this represents a gap of approximately 2 3 db from the shannon limit of the additive white gaussian noise channel which could be closed to within 0 8 1 2 db using constellation shaping

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

on the spectral moments of unicyclic graphs with fixed diameter abstract unicyclic graphs are connected graphs in which the number of edges equals the number of vertices let u n d be the class of unicyclic graphs of order n and diameter d for unicyclic graphs lexicographic ordering by spectral moments s order is discussed in this paper the last d d 2 2 graphs in the s order among all unicyclic graphs in u n d 3 d n 5 are characterized in addition for the cases d n 4 and d n 3 the last 2 n 11 and last n n 2 7 unicyclic graphs in the set u n d are also given respectively

on the spectral radius of tricyclic graphs with a maximum matching abstract in this paper we determine the graph with the largest spectral radius among all the tricyclic graphs with a maximum matching of cardinality m m 8

on the spectral radius of bicyclic graphs with n vertices and diameter d bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one in this paper we determine the graph with the largest spectral radius among all bicyclic graphs with n vertices and diameter d as an application we give first three graphs among all bicyclic graphs on n vertices ordered according to their spectral radii in decreasing order

an efficient block based addressing scheme for the nearly optimum shaping of multidimensional signal spaces we introduce an efficient addressing scheme for the nearly optimum shaping of a multidimensional signal constellation the 2 d two dimensional subspaces are partitioned into k energy shells of equal cardinality the average energy of a 2 d shell can be closely approximated by a linear function of its index in an n 2n d space we obtain k sup n shaping clusters of equal cardinality shaping is achieved by selecting t spl les k sup n of the n d clusters with the least sum of the 2 d indices this results in a set of t integer n tuples with the components in the range 0 k 1 and the sum of the components being at most a given number l the problem of addressing is to find a one to one mapping between the set of such n tuples and the set of integers 0 t 1 such that the mapping and its inverse can be easily implemented in the proposed scheme the n d clusters are grouped into blocks of identical binary weight vectors this results in a simple rule for the addressing of points within the blocks the addressing of the blocks is based on some recursive relationship which allows us to decompose the problem into simpler parts the overall scheme requires a modest amount of memory and has a small computational complexity

ordering trees by their largest eigenvalues abstract the set of trees with n vertices is denoted by t n hofmeister has determined the first five values of the largest eigenvalue of trees in t n and the corresponding trees for these values linear algebra appl 260 1997 43 in other words an order of the first five trees in t n by their largest eigenvalues has been given focus on the same purpose we shall give a partition for trees in t n first in the this paper and then extend this order to the eighth tree

on the spectral radius of quasi tree graphs abstract a connected graph g v e is called a quasi tree if there exists u 0 v g such that g u 0 is a tree denote q n d 0 g g is a quasi tree graph of order n with g u 0 being a tree and d g u 0 d 0 in this paper we determined the maximal and the second maximal spectral radii of all quasi tree graphs in the set q n d 0

shaping multidimensional signal spaces ii shell addressed constellations for pt i see ibid pp 1799 1808 nov 1993 by appropriately selecting the boundary of a multidimensional signal constellation used for data transmission the average energy of the constellation can be reduced reduction in the average energy shaping gain is obtained at the price of increasing the constellation expansion ratio cer sub s and the peak to average power ratio par the authors describe some practical means for selecting the boundary so as to achieve a point with low addressing complexity near the knee of the corresponding tradeoff curves shaping gain versus cer sub s or par one class of addressing schemes is based on using a lookup table a method to facilitate the realization of the addressing lookup table is introduced this method is based on the decomposition of addressing into a hierarchy of addressing steps each of a low complexity this avoids exponential growth of the complexity using this addressing decomposition and a memory of a practical size one can move along a tradeoff curve which has negligible suboptimality another class of addressing schemes is based on using a voronoi constellation in a space of half the original dimensionality

optimal shaping properties of the truncated polydisc multidimensional constellation shaping with a family of regions called truncated polydiscs is studied this family achieves maximum shaping gain for a given two dimensional peak to average energy ratio or a given two dimensional constellation expansion ratio an efficient algorithm for mapping data words to constellation points is described that requires o n log n arithmetic operations and o n sup 2 lookup table space truncated polydisc shaping can easily be incorporated into standard coded modulation schemes

multidimensional signal sets through the shell construction for parallel channels the authors derive a procedure to send r bits on m parallel channels a decomposition of the best constellation in z sup m 1 2 1 2 is given in terms of the cross products of lower dimensional shells of points the proposed scheme can be used with good known coset codes to provide an alternate method of coded modulation the results indicate that one can get good shaping gains for low encoder complexity the method is also generalized for channels with unequal gains the authors also find a significant performance advantage at fixed shaping gain in certain cases with respect to the recent voronoi constellations in terms of peak to average power and constellation expansion

trellis shaping the author discusses trellis shaping a method of selecting a minimum weight sequence from an equivalence class of possible transmitted sequences by a search through the trellis diagram of a shaping convolutional code c sub s shaping gains on the order of 1 db may be obtained with simple four state shaping codes and with moderate constellation expansion the shaping gains obtained with more complicated codes approach the ultimate shaping gain of 1 53 db with a feedback free syndrome former for c sub s transmitted data can be recovered without catastrophic error propagation constellation expansion and peak to average energy ratio may be effectively limited by peak constraints with lattice theoretic constellations the shaping operation may be characterized as a decoding of an initial sequence in a channel trellis code by a minimum distance decoder for a shaping trellis code based on the shaping convolutional code and the set of possible transmitted sequences is then the set of code sequences in the channel trellis code that lie in the voronoi region of the trellis shaping code

on the two largest eigenvalues of trees abstract very little is known about upper bounds for the largest eigenvalues of a tree that depend only on the vertex number starting from a classical upper bound for the largest eigenvalue some refinements can be obtained by successively removing trees from consideration the results can be used to characterize those trees that maximize the second largest eigenvalue this corrects a result from the literature and it includes a proof of a conjecture of neumaier the main tool for this endeavor is the theory of partial engenvectors

on the index of tricyclic graphs with perfect matchings abstract let t 2 k be the set of all tricyclic graphs on 2 k k 2 vertices with perfect matchings in this paper we discuss some properties of the connected graphs with perfect matchings and then determine graphs with the largest index in t 2 k

some notes on graphs whose index is close to 2 abstract we consider two classes of graphs i trees of order n and diameter d n 3 and ii unicyclic graphs of order n and girth g n 2 assuming that each graph within these classes has two vertices of degree 3 at distance k we order by the index i e spectral radius the graphs from i for any fixed k 1 k d 2 and the graphs from ii independently of k

bandwidth efficient and rate matched low density parity check coded modulation a new coded modulation scheme is proposed at the transmitter the concatenation of a distribution matcher and a systematic binary encoder performs probabilistic signal shaping and channel coding at the receiver the output of a bitwise demapper is fed to a binary decoder no iterative demapping is performed rate adaption is achieved by adjusting the input distribution and the transmission power the scheme is applied to bipolar amplitude shift keying ask constellations with equidistant signal points and it is directly applicable to two dimensional quadrature amplitude modulation qam the scheme is implemented by using the dvb s2 low density parity check ldpc codes at a frame error rate of 10 3 the new scheme operates within less than 1 1 db of the awgn capacity frac 1 2 log 2 1 mathsf snr at any spectral efficiency between 1 and 5 bits s hz by using only 5 modes i e 4 ask with code rate 2 3 8 ask with 3 4 16 ask and 32 ask with 5 6 and 64 ask with 9 10

on the spectral radius of unicyclic graphs with fixed diameter in the paper we will determine graphs with the maximal spectral radius among all the unicyclic graphs with n vertices and diameter d

reducibility of certain induced representations of e6 and e7 abstract we give equivalent conditions for certain parabolically induced representations of e 6 and e 7 to be irreducible by calculating the action of certain weyl group element and detecting poles of the first two langlands shahidi l functions at s 0

on the spectral radius of trees abstract in this paper we investigated the spectral radius of trees and obtained the following result let t be a tree on n vertices let m t denote one maximum matching of t and m t i let t i be a tree as shown in fig 1 then t 1 2 n i 1 n i 1 2 4 n 2i 1 and equality holds if and only if t t i