on the k tuple domination of generalized de brujin and kautz digraphs a vertex u in a digraph g v a is said to dominate itself and vertices v such that u v a for a positive integer k a k tuple dominating set of g is a subset d of vertices such that every vertex in g is dominated by at least k vertices in d the k tuple domination number of g is the minimum cardinality of a k tuple dominating set of g this paper deals with the k tuple domination problem on generalized de bruijn and kautz digraphs we establish bounds on the k tuple domination number for the generalized de bruijn and kautz digraphs and we obtain some conditions for the k tuple domination number attaining the bounds

spin holonomy of einstein manifolds berger s theorem classifies the linear holonomy groups of irreducible simply connected riemannian manifolds for physical applications however it is at least as important to have a classification of the possible spin holonomy groups defined by the parallel transport of spinors of non simply connected manifolds we establish a complete classification of the spin holonomy groups of all compact locally irreducible einstein riemannian spin manifolds of non negative scalar curvature

the map between conformal hypercomplex hyper kahler and quaternionic kahler geometry we review the general properties of target spaces of hypermultiplets which are quaternionic like manifolds and discuss the relations between these manifolds and their symmetry generators we explicitly construct a one to one map between conformal hypercomplex manifolds i e those that have a closed homothetic killing vector and quaternionic manifolds of one quaternionic dimension less an important role is played by transformations relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds in this map the subclass of conformal hyper kahler manifolds is mapped to quaternionic kahler manifolds we relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other

some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

on twistor transformations and invariant differential operator of simple lie group g2 2 the twistor transformations associated to the simple lie group g2 are described explicitly we consider the double fibration g2 p2 g2 b g2 p1 where p1 and p2 are two parabolic subgroups of g2 and b is a borel subgroup and its local version h2 f h1 where h1 is the heisenberg group of dimension 5 embedded in the coset space g2 p1 f cp1 h1 and h2 contains the nilpotent lie group h2 of step three the baker campbell hausdorff formula is used to parametrize the coset spaces coordinates charts their transition functions and the fibers of the projection as complex curves we write down the relative de rham sequence on f along the fibers and push it down to h1 to get a family of matrix valued differential operators dk then we establish a kind of penrose correspondence for g2 the kernel of dk is isomorphic to the first cohomology of the sheaf o k over h2 we also give the penrose type integral transformation u pf for f o k which gives solutions to equations dku 0 when restricted to the re

minimal representations geometric quantization and unitarity in the framework of geometric quantization we explicitly construct in a uniform fashion a unitary minimal representation pio of every simply connected real lie group go such that the maximal compact subgroup of go has finite center and go admits some minimal representation we obtain algebraic and analytic results about pio we give several results on the algebraic and symplectic geometry of the minimal nilpotent orbits and then quantize these results to obtain the corresponding representations we assume lie go c is simple

special prehomogeneous vector spaces associated to f4 e6 e7 e8 and simple jordan algebras of rank 3 abstract a simple complex lie algebra has a natural grading g g 2 g 1 g 0 g 1 g 2 associated to the highest root it is related to the quaternionic real form of g the real rank of the associated symmetric space is shown to be at most 4 when the rank is equal to 4 i e g a n c n g 2 the semi simple lie algebra m g 0 g 0 is shown to be the conformal algebra of a rank 3 semi simple jordan algebra v if g is one of the four exceptional lie algebras f4 e6 e7 e8 then v is simple the vector space g 1 is a prehomogeneous vector space under the action of m c where m int m is the adjoint group of m it admits a non zero m invariant polynomial of degree 4 conversely to any simple jordan algebra v of rank 3 is naturally associated a representation of a twofold covering of its conformal group co v on a vector space w such that w is a prehomogeneous vector space under the action of co v c an invariant polynomial of degree 4 is explicitly constructed a geometric description of all the orbits of co v in w is given

on certain distinguished unitary representations supported on null cones let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i or inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i and let g u p q inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i be the isometry group of a inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i hermitian form of signature p q for 2 n min p q we consider the action of g on v n the direct sum of n copies of the standard module v inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i and the associated action of g on the regular part of the null cone denoted by x 00 we show that there is a commuting set of g invariant differential operators acting on the space of c functions on x 00 which transform according to a distinguished gl n inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i character and the resulting kernel is an irreducible unitary representation of g our result can be interpreted as providing a geometric construction of the theta lift of the characters from the group g u n n or o 4 n the construction and approach here follow a previous work of zhu and huang representation theory 1 1997 where the group concerned is g o p q with p q even

restrictions of quaternionic representations in a paper by gross and wallach 1996 j reine angew math 481 73 123 the k types of the continuations of the quaternionic discrete series of a quaternionic lie group g are associated with projective orbits o of certain subgroups in g c in this paper we will show that the restrictions of the representations to quaternionic subgroups are closely related with the intersection of the zariski closure of o with hyperplanes we apply this to the minimal representations of the exceptional groups of real rank 4 and investigate the correspondences of certain compact dual pairs

compact 8 manifolds with holonomy spin 7 in berger s classification 4 of the possible holonomy groups of a nonsymmetric irreducible riemannian manifold there are two special cases the exceptional holonomy groups g2 in 7 dimensions and spin 7 in 8 dimensions bryant 6 proved that such metrics exist locally using the theory of exterior differential systems and gave some explicit examples later bryant and salamon 7 constructed explicit complete metrics with holonomy g2 and spin 7 in two previous papers 12 13 the author constructed many examples of compact riemannian 7 manifolds with holonomy g2 this paper will construct examples of compact riemannian 8 manifolds with holonomy spin 7 using similar methods we believe that these are the first examples known since metrics with holonomy spin 7 are ricci flat these are also new examples of compact ricci flat riemannian 8 manifolds let m be an 8 manifold a spin 7 structure on m can be encoded in a 4 form on m a special 4 form satisfying the condition that the stabilizer of at each point should be isomorphic to spin 7 by an abuse of notation we usually identify the spin 7 structure with its associated 4 form since spin 7 so 8 the spin 7 structure also induces a riemannian metric g and an orientation on m it turns out that the holonomy group hol g of g is a subgroup of spin 7 with spin 7 structure if and only if d 0 the quantity d is called the torsion of the spin 7 structure and is called torsion free if d 0 very briefly the plan of the paper splits into four steps as follows firstly a compact 8 manifold m is given secondly a spin 7 structure on m is found with small torsion i e d is small thirdly is deformed to a nearby spin 7 structure with d 0 thus is torsion free and if g is the associated metric then hol g spin 7 fourthly it is shown that hol g is spin 7 and not some proper subgroup the structure of the paper is designed around this division of the construction into four steps there are six chapters this first chapter is of introductory material the second chapter has the full statements of the main results in

classification of minimal representations of real simple lie groups based on an idea by gan and savin represent theory 9 46 93 2005 we give a classification of minimal representations of connected simple real lie groups not of type a actually we prove that there exist no new minimal representations up to infinitesimal equivalence

realization of quaternionic discrete series on the unit ball in h d abstract we give a realization for the quaternionic discrete series for the group sp 1 d as the kernel of certain elementary cauchy riemann fueter operator and we compute the corresponding reproducing kernels

quasi conformal actions quaternionic discrete series and twistors su 2 1 and g 2 2 quasi conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane to hermitian symmetric tube domains based on arbitrary jordan algebras and further to arbitrary freudenthal triple systems in the mathematics literature quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionic kahler spaces these two constructions are essentially identical as we show explicitly for the two rank 2 cases su 2 1 and g2 2 we obtain explicit results for certain principal series quaternionic discrete series and minimal representations of these groups including formulas for the lowest k types in various polarizations we expect our results to have applications to topological strings black hole micro state counting and to the theory of automorphic forms

fourier jacobi expansion of automorphic forms on sp 1 q generating quaternionic discrete series abstract the aim of this paper is to develop the notion of the fourier expansion of automorphic forms on sp 1 q generating quaternionic discrete series which are non holomorphic forms there is such an expansion given by tsuneo arakawa assuming the boundedness of the forms and the integrability of the discrete series we study these automorphic forms without such assumptions when q 1 we prove the koecher principle for such automorphic forms whose validity is known for holomorphic automorphic forms except elliptic modular forms

nilpotent orbits and some small unitary representations of indefinite orthogonal groups abstract for 2 m l 2 let g be a simply connected lie group with g 0 so 2m 2l 2m as lie algebra let g k p be the complexification of the usual cartan decomposition let k be the analytic subgroup with lie algebra k g 0 and let u g be the universal enveloping algebra of g this work examines the unitarity and k spectrum of representations in the analytic continuation of discrete series of g relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g the roots with respect to the usual compact cartan subalgebra are all ei ej with 1 i q l u be the parabolic subalgebra of g for which em em 1 contributes to u and the other simple roots contribute to l let l be the analytic subgroup of g with lie algebra l g 0 let l c int g l let 2 u be the sum of the roots contributing to u and let q l u be the parabolic subalgebra opposite to q the members of u p are nilpotent members of g the group l c acts on u p with finitely many orbits and the topological closure of each orbit is an irreducible algebraic variety if y is one of these varieties let r y be the dual coordinate ring of y this is a quotient of the algebra of symmetric tensors on u p that carries a fully reducible representation of l c for s z let s k 1 m l s 2 e k then s defines a one dimensional l l module c s extend this to a q l module by having u act by 0 and define n s 2 u u g q c s 2 u let n s 2 u be the unique irreducible quotient of n s 2 u the representations under study are s s n s 2 u and s s n s 2 u where s dim u k and s is the sth derived bernstein functor for s 2l 2 it is known that s s and that s is in the discrete series enright parthsarathy wallach and wolf showed for m s 2l 2 that s s and that s is still unitary the present paper shows that s is unitary for 0 s m 1 even though s s and it relates the k spectrum of the representations s to the representation of l c on a suitable r y with y depending on s use of a branching formula of d e littlewood allows one to obtain an explicit multiplicity formula for each k type in s the variety y is indispensable in the proof the chief tools involved are an idea of b gross and wallach a geometric interpretation of littlewood s theorem and some estimates of norms it is shown further that the natural invariant hermitian form on s does not make s unitary for s l c on any r y a final section of the paper treats in similar fashion the simply connected lie group with lie algebra g 0 so 2m 2l 2m 1 2 m l 2

a notion of rank for unitary representations of reductive groups based on kirillov s orbit method we introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero the theory is based on kirillov s method of orbits for nilpotent groups over local fields when the semisimple group is a classical group we prove that the new theory is essentially equivalent to howe s theory of n rank see ho4 l2 sc therefore our results provide a systematic generalization of the notion of a small representation in the sense of howe to exceptional groups however unlike previous works that used ad hoc methods to study different types of classical groups and some exceptional ones see we ls our definition is simultaneously applicable to both classical and exceptional groups the most important result of this article is a general purity result for unitary representations which demonstrates how similar partial results in these authors works should be formulated and proved for an arbitrary semisimple group in the language of kirillov s theory the purity result is a crucial step toward studying small representations of exceptional groups new results concerning small unitary representations of exceptional groups will be published in a forthcoming paper s

minimal representations spherical vectors and exceptional theta series theta series for exceptional groups have been suggested as a possible description of the eleven dimensional quantum supermembrane we present explicit formulae for these automorphic forms whenever the underlying lie group g is split or complex and simply laced specifically we review and construct explicitly the minimal representation of g generalizing the schrodinger representation of symplectic groups we compute the spherical vector in this representation i e the wave function invariant under the maximal compact subgroup which plays the role of the summand in the automorphic theta series we also determine the spherical vector over the complex field we outline how the spherical vector over the p adic number fields provides the summation measure in the theta series postponing its determination to a sequel of this work the simplicity of our result is suggestive of a new born infeld like description of the membrane where u duality is realized non linearly our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries

analysis on the minimal representation of o p q abstract for the group o p q we give a new construction of its minimal unitary representation via euclidean fourier analysis this is an extension of the q 2 case where the representation is the mass zero spin zero representation realized in a hilbert space of solutions to the wave equation the group o p q acts as the mobius group of conformal transformations on r p 1 q 1 and preserves a space of solutions of the ultrahyperbolic laplace equation on r p 1 q 1 we construct in an intrinsic and natural way a hilbert space of solutions so that o p q becomes a continuous irreducible unitary representation in this hilbert space we also prove that this representation is unitarily equivalent to the representation on l2 c where c is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the lie algebra of o p q

the bottleneck independent domination on the classes of bipartite graphs and block graphs let g v e w denote a graph with vertex set v and edge set e and each vertex v is associated with a cost w v for any set v v the bottleneck cost of v is defined as max w x x v this paper considers the bottleneck independent dominating set problem the bids problem which determines an independent dominating set of g such that its bottleneck cost is minimized this paper studies the problem on the classes of bipartite graphs and block graphs this paper first proves that the problem is np hard on chordal bipartite graphs second a linear time algorithm on convex bipartite graphs is proposed next a linear time algorithm on trees is designed then we generalize the algorithmic result to block graphs all algorithms are designed by the dynamic programming strategy

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

on rainbow domination numbers of graphs given a graph g and a set of t colors assume that we assign an arbitrary subset of these colors to each vertex of g if we require that each vertex to which an empty set is assigned has in its neighborhood all t colors then this assignment is called a t rainbow dominating function of the graph g the corresponding invariant c r t g which is the minimum sum of numbers of assigned colors over all vertices of g is called the t rainbow domination number of g in this paper bounds for the t rainbow domination number of an arbitrary graph for an arbitrary positive integer t are given the 3 rainbow domination numbers of several classes of graphs such as paths cycles and the generalized petersen graphs p n k are investigated the 3 rainbow domination number of p n 1 is determined and the upper bounds for p n 2 and p n 3 are provided by computer search we obtain that the upper bounds for p n 2 match their exact values for n 17

conformal and quasiconformal realizations of exceptional lie groups we present a nonlinear realization of e8 8 on a space of 57 dimensions which is quasiconformal in the sense that it leaves invariant a suitably defined light cone in 57 this realization which is related to the freudenthal triple system associated with the unique exceptional jordan algebra over the split octonions contains previous conformal realizations of the lower rank exceptional lie groups on generalized space times and in particular a conformal realization of e7 7 on 27 which we exhibit explicitly possible applications of our results to supergravity and m theory are briefly mentioned

d4 modular forms in this paper we study modular forms on two simply connected groups of type d 4 over q one group g s is a globally split group of type d 4 viewed as the group of isotopies of the split rational octonions the other g c is the isotopy group of the rational nonsplit octonions we study automorphic forms on g s in analogy to the work of gross gan and savin on g 2 namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation we study automorphic forms on g c using gross s formalism of algebraic modular forms finally we follow work of gan savin gross rallis and others to study an exceptional theta correspondence connecting modular forms on g c and g s this can be thought of as an octonionic generalization of the jacquet langlands correspondence

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

efficient algorithms for the minimum connected domination on trapezoid graphs given the trapezoid diagram the problem of finding the minimum cardinality connected dominating set in trapezoid graphs was solved in o m n time y d liang steiner set and connected domination in trapezoid graphs inform process lett 56 2 1995 101 108 kohler e kohler connected domination and dominating clique in trapezoid graphs discr appl math 99 2000 91 110 recently improved this result to o n time for the vertex weighted case the problem of finding the minimum weighted connected dominating set in trapezoid graphs can be solved in o m nlogn time anand srinivasan m s chang k madhukar c pandu rangan efficient algorithms for the weighted domination problems on trapezoid graphs manuscript 1996 herein n m denotes the number of vertices edges of the trapezoid graph in this paper we show a different approach for the problem of finding the minimum cardinality connected dominating set in trapezoid graphs using o n time for finding the minimum weighted connected dominating set we show the problem can be solved in o nloglogn time

rainbow domination numbers of generalized petersen graphs abstract domination and its variations in graphs are natural models for the location problems in operations research in this paper we investigate the rainbow domination number of graphs which was introduced by bresar henning and rall given a graph g and a positive integer t a t rainbow dominating function of g is a function f from vertex set to the set of all subsets of 1 2 t such that for any vertex v with f v we have u n v f u 1 2 t the t rainbow domination problem is to determine the t rainbow domination number rt g of g that is the minimum value of v v g f v where f runs over all t rainbow dominating functions of g the domination number and its variations of generalized petersen graphs p n k are widely investigated the exact values of r2 p n 1 and r3 p n 1 are already determined in 11 12 in this paper we determine the exact values of rt p n 1 for any t 8 and t 4 and prove that r t p 2 k k 4 k for t 6 where p 2k k is a special graph

jordan algebras exceptional groups and bhargava composition abstract we consider an integral version of the freudenthal construction relating jordan algebras and exceptional algebraic groups we show how this construction is related to higher composition laws of m bhargava in number theory we propose an algorithmic approach to studying orbit spaces of groups underlying bhargava s higher composition laws using this method we discover two new examples of spaces sharing similar properties and indicate several more examples of spaces where such composition laws may be introduced

theta lifting from elliptic cusp forms to automorphic forms on sp 1 q tsuneo arakawa formulated a theta lifting from elliptic cusp forms to automorphic forms on sp 1 q in his unpublished note which was inspired by kudla lifting i e a theta lifting from elliptic modular forms to holomorphic automorphic forms on su 1 q we prove that the images of arakawa s theta lifting belong to the space of bounded automorphic forms generating quaternionic discrete series which are non holomorphic forms in the appendix we provide the construction of eisenstein series and poincare series generating such discrete series

the spectral function for sturm liouville problems where the potential is of wigner von neumann type or slowly decaying we consider the linear second order differential equation y q x y 0on 0 with the boundary condition y 0 cos y 0 sin 0forsome 0 we suppose that q x is real valued continuously differentiable and that q x 0 as x with q l1 0 our main object of study is the spectral function associated with and we derive a series expansion for this function valid for 0 where 0 is computable and establish a 1 also computable such that and with 0 have no points of spectral concentration for 1 we illustrate our results with examples in particular we consider the case of the wigner von neumann potential

dominating sets in directed graphs we consider the problem of incrementally computing a minimal dominating set of a directed graph after the insertion or deletion of a set of arcs earlier results have either focused on the study of the properties that minimum not minimal dominating sets preserved or lacked to investigate which update affects a minimal dominating set and in what ways in this paper we first show how to incrementally compute a minimal dominating set on arc insertions we then reduce the case of computing a minimal dominating set on arc deletions to the case of insertions some properties on minimal dominating sets are provided to support the incremental strategy lastly we give a new bound on the size of minimum dominating sets based on those results

special metrics and triality abstract we investigate a new 8 dimensional riemannian geometry defined by a generic closed and coclosed 3 form with stabiliser psu 3 and which arises as a critical point of hitchin s variational principle we give a riemannian characterisation of this structure in terms of invariant spinor valued 1 forms which are harmonic with respect to the twisted dirac operator on 1 we establish various obstructions to the existence of topological reductions to psu 3 for compact manifolds we also give sufficient conditions for topological psu 3 structures that can be lifted to topological su 3 structures we also construct the first known compact example of an integrable non symmetric psu 3 structure in the same vein we give a new riemannian characterisation for topological quaternionic kahler structures which are defined by an sp 1 sp 2 invariant self dual 4 form again we show that this form is closed if and only if the corresponding spinor valued 1 form is harmonic for and that these equivalent conditions produce constraints on the ricci tensor

dual pair correspondences for non linear covers of orthogonal groups abstract in this paper we study compact dual pair correspondences arising from smallest representations of non linear covers of odd orthogonal groups we identify representations appearing in these correspondences with subquotients of cohomologically induced representations

the sextonions and e712 we fill in the hole in the exceptional series of lie algebras that was observed by cvitanovic deligne cohen and deman more precisely we show that the intermediate lie algebra between e7 and e8 satisfies some of the decomposition and dimension formulas of the exceptional simple lie algebras a key role is played by the sextonions a six dimensional algebra between the quaternions and octonions using the sextonions we show similar results hold for the rows of an expanded freudenthal magic chart we also obtain new interpretations of the adjoint variety of the exceptional group g2

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained

dominating problems in swapped networks swapped networks sns are a family of two level interconnection networks suitable for constructing large parallel and distributed systems in this paper the minimum dominating set mds problem and the minimum connected dominating set mcds problem in sns are investigated based on the connectivity rule of sns we prove the two problems in sns are np hard and present two efficient algorithms for building dominating sets and connected dominating sets in sns the proposed algorithms use as input a given connected dominating set of the factor network and yield a good approximation of an mds or mcds for the sn provided that the input is a good approximation of an mds or mcds for the factor network we also derive several non trivial bounds on the connected domination parameters of sns we believe this work is of theoretical interest in graph theory since sns form a family of graphs it may also motivate further research on dominating problems in sns with their potential applications