an augmented electric field integral equation for layered medium green s function this paper proposes an augmented electric field integral equation a efie for layered medium green s function the newly developed matrix friendly formulation of layered medium green s function is applied in this method by separating charge as extra unknown list and enforcing the current continuity equation the traditional efie can be cast into a generalized saddle point system frequency scaling for the matrix friendly formulation is analyzed when frequency tends to zero rank deficiency and the charge neutrality enforcement of the a efie for layered medium green s function is discussed in detail the electrostatic limit of the a efie is also analyzed without any topological loop searching algorithm electrically small conducting structures embedded in a general layered medium can be simulated by using this new a efie formulation several numerical results are presented to validate this method at the end of this paper

q analogs of distance matrices of 3 hypertrees abstract we consider the distance matrix of trees in 3 uniform hypergraphs which we call 3 hypertrees we give a formula for the inverse of a few q analogs of distance matrices of 3 hypertrees t some results are analogs of results by bapat et al for graphs we give an alternate proof of the result that the determinant of the distance matrix of a 3 hypertree t depends only on n the number of vertices of t further we give a pfaffian identity for a principal submatrix of some skew symmetrized distance matrices of 3 hypertrees when we fix an ordering of the vertices and assign signs appropriately a result of graham hoffman and hosoya relates the determinant of the distance matrix of a graph and the determinants of its two connected blocks when the graph has as blocks a fixed connected graph h which satisfy some conditions we give a formula for the inverse of its distance matrix this result generalises a result of graham and lovasz when each block of g is a fixed graph g we also give some corollaries about the sum of the entries of the inverse of the distance matrix of g and some of its analogs

resistance matrices of graphs with matrix weights abstract the resistance matrix of a simple connected graph g is denoted by r and is defined by r r i j where r i j is the resistance distance between the vertices i and j of g in this paper we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size we derive a formula for the determinant and the inverse of the resistance matrix then we establish an interlacing inequality for the eigenvalues of resistance and laplacian matrices of tree using this interlacing inequality we obtain the inertia of the resistance matrix of tree

two laplacians for the distance matrix of a graph abstract we introduce a laplacian and a signless laplacian for the distance matrix of a connected graph called the distance laplacian and distance signless laplacian respectively we show the equivalence between the distance signless laplacian distance laplacian and the distance spectra for the class of transmission regular graphs there is also an equivalence between the laplacian spectrum and the distance laplacian spectrum of any connected graph of diameter 2 similarities between n as a distance laplacian eigenvalue and the algebraic connectivity are established

sofic groups and direct finiteness abstract we construct an analogue of von neumann s affiliated algebras for sofic group algebras over arbitrary fields consequently we settle kaplansky s direct finiteness conjecture for sofic groups

sofic groups and diophantine approximation we prove the algebraic eigenvalue conjecture of j dodziuk p linnell v mathai t schick and s yates see 2 for sofic groups moreover we give restrictions on the spectral measure of elements in the integral group ring finally we define integer operators and prove a quantization of the operator norm below 2 to the knowledge of the author there is no group known that is not sofic

an accurate scheme for the solution of the time domain integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation despite the numerous advances made in increasing the computational efficiency of time domain integral equation tdie based solvers the stability and accuracy of tdie solvers remain problematic this paper introduces a new numerical method for the accurate solution of tdies for scattering from arbitrary perfectly conducting surfaces the work described in this paper uses the higher order divergence conforming basis functions of graglia et al for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations since the basis functions used for the temporal representation are noncausal an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization

investigation of a boundary integral equation mbi n times mbi h mbi j _ s on torus shaped perfect conductors we have found that a boundary integral equation n times h js results in an inaccurate solution on torus shaped perfect conducting surfaces although constraining a boundary condition n times h js automatically satisfies e x n 0 on a closed polyhedron shaped conductor there exists a solution which only satisfies the boundary condition n times h js but not the other boundary condition e times n 0 on a torus shaped conductor we have introduced a virtual magnetic current ms in the system equation as another degree of freedom and employed e times n 0 at one point on the perfect conducting surface so as to obtain the accurate solution to verify the proposed discussion we have formulated an axially symmetric method of moments based on n times h js and compared with other electromagnetic field solvers

inverses of q distance matrices of a tree the determinant and the inverse of the distance matrix of a tree have been investigated in the literature following the classical formulas due to graham and pollak for the determinant and due to graham and lovasz for the inverse we consider two q analogs of the distance matrix of a tree and obtain formulas for the inverses of the two distance matrices yan and yeh have previously obtained expressions for the determinants of the two distance matrices some related results are proved

inverse of the distance matrix of a cactoid digraph abstract a strongly connected directed graph in which any two directed cycles of it share at most one common vertex is called a cactoid digraph let d be its distance matrix by a theorem of graham hoffman and hosoya we have det d 0 in this paper we give the formulas for both the determinant and the inverse of the distance matrix d for a cactoid digraph

on cartesian product of euclidean distance matrices abstract if a r m m and b r n n we define the product a b as a b a j n j m b where denotes the kronecker product and j n is the n n matrix of all ones we refer to this product as the cartesian product of a and b since if d 1 and d 2 are the distance matrices of graphs g 1 and g 2 respectively then d 1 d 2 is the distance matrix of the cartesian product g 1 g 2 we study cartesian products of euclidean distance matrices edms we prove that if a and b are edms then so is the product a b we show that if a is an edm and u is symmetric then a u is an edm if and only if u c j n for some c it is shown that for edms a and b a b is spherical if and only if both a and b are spherical if a and b are edms then we derive expressions for the rank and the moore penrose inverse of a b in the final section we consider the product a b for arbitrary matrices for a r m m b r n n we show that all nonzero minors of a b of order m n 1 are equal an explicit formula for a nonzero minor of order m n 1 is proved the result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n

a finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics a new method for solving the time domain integral equations of electromagnetic scattering from conductors is introduced this method called finite difference delay modeling appears to be completely stable and accurate when applied to arbitrary structures the temporal discretization used is based on finite differences specifically based on a mapping from the laplace domain to the z transform domain first and second order unconditionally stable methods are derived spatial convergence is achieved using the higher order divergence conforming vector bases of graglia et al low frequency instability problems are avoided with the loop tree decomposition approach numerical results will illustrate the accuracy and stability of the technique

a surface integral equation formulation for low frequency scattering from a composite object a surface integral equation formulation is derived for low frequency scattering from a composite object this formulation allows those algorithms for finding the loop tree basis on simple surfaces to be applied to the complicated interfaces of a composite object by including the residue term in the k operator the present formulation induces the interface boundary conditions automatically and the resultant matrix equation can be solved directly without using the o n sup 2 number of unknowns reduction scheme numerical results validate that the algorithm is stable even at extremely low frequencies and for closely spaced structures

distance spectral radius of graphs with r pendent vertices abstract let g n r be the class of all connected graphs of order n with r pendent vertices in this paper we determine the unique graph with minimal distance spectral radius in g n r in addition we determine the unique graph with maximal distance spectral radius in g n r for each r 2 3 n 3 n 2 n 1

squared distance matrix of a tree inverse and inertia abstract let t be a tree with vertices v t 1 n the distance between vertices i j v t denoted d i j is defined to be the length the number of edges of the path from i to j we set d i i 0 i 1 n the squared distance matrix of t is the n n matrix with i j element equal to 0 if i j and d i j 2 if i j it is known that is nonsingular if and only if the tree has at most one vertex of degree 2 we obtain a formula for 1 if it exists when the tree has no vertex of degree 2 the formula is particularly simple and depends on a certain two step laplacian of the tree we determine the inertia of the inverse and the inertia of the edge orientation matrix are also described

proof of a conjecture involving the second largest d eigenvalue and the number of triangles abstract let g be a connected graph of order n with tr triangles and d be the distance matrix of g let 1 d 2 d n d be the d eigenvalue of the graph g fajtlowicz 1998 4 conjectured that 2 d tr when the independent number g 2 in this paper the conjecture is confirmed and the extremal graph when the equality holds is characterized

identities for minors of the laplacian resistance and distance matrices it is shown that if l and d are the laplacian and the distance matrix of a tree respectively then any minor of the laplacian equals the sum of the cofactors of the complementary submatrix of d up to sign and a power of 2 an analogous more general result is proved for the laplacian and the resistance matrix of any graph a similar identity is proved for graphs in which each block is a complete graph on r vertices and for q analogues of such matrices of a tree our main tool is an identity for the minors of a matrix and its inverse

the singular function expansion in time dependent scattering a class of linear initial boundary value problems relevant to time dependent scattering is solved exactly in terms of the complete orthogonal set of singular functions of the appropriate integral operator the method is primarily presented as a mathematically defensible alternative to the eigenmode expansion method the results extend the singularity expansion method to include the steady state response at source frequencies and provide a connection between resonance behavior in the steady state and low loss natural modes the method furthermore leads to an efficient automatic algorithm for the determination of natural modes and natural frequencies by observation of the singular values the conclusions are supported by numerical calculations

extremal cacti of given matching number with respect to the distance spectral radius a cactus is a connected graph in which any two cycles have at most one common vertex the distance spectral radius g of a graph g is the largest eigenvalue of the distance matrix d g recently many researchers proposed the use of g as a molecular structure descriptor of alkanes in this paper we characterize n vertex cyclic cactus with given matching number m which minimizes the distance spectral radius the resulting cactus also minimizes the hosoya index the wiener index and the randic index in the same class of graphs

convergence and limits of linear representations of finite groups motivated by the theory of graph limits we introduce and study the convergence and limits of linear representations of finite groups over finite fields the limit objects are infinite dimensional representations of free groups in continuous algebras we show that under a certain integrality condition the algebras above are skew fields our main result is the extension of schramm s characterization of hyperfiniteness to linear representations

the product distance matrix of a tree with matrix weights on its arcs abstract let t be a tree with vertex set n 1 2 n for each i n let m i be a positive integer an ordered pair of two adjacent vertices is called an arc each arc i j of t has a weight w i j which is an m i m j matrix for two vertices i j n let the unique directed path from i to j be p i j x 0 x 1 x d where d 1 x 0 i and x d j define the product distance from i to j to be the m i m j matrix m i j w x 0 x 1 w x 1 x 2 w x d 1 x d let n i 1 n m i the n n product distance matrix d of t is a partitioned matrix whose i j block is the matrix m i j we give a formula for det d when det d 0 the inverse of d is also obtained these generalize known results for the product distance matrix when either the weights are real numbers or m 1 m 2 m n s and the weights w i j w j i w e for each edge e i j e t

on distance matrices and laplacians we consider distance matrices of certain graphs and of points chosen in a rectangular grid formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved if d is the distance matrix of a tree then we obtain certain results for a perturbation of d 1 as an example it is shown that if l is the laplacian matrix of an arbitrary connected graph then d 1 l 1 is an entrywise positive matrix we consider the distance matrix of a subset of a rectangular grid of points in the plane if we choose m k 1 points not containing a closed path in an m k grid then a formula for the determinant of the distance matrix of such points is obtained

on euclidean distance matrices abstract if a is a real symmetric matrix and p is an orthogonal projection onto a hyperplane then we derive a formula for the moore penrose inverse of pap as an application we obtain a formula for the moore penrose inverse of an euclidean distance matrix edm which generalizes formulae for the inverse of a edm in the literature to an invertible spherical edm we associate a laplacian matrix which we define as a positive semidefinite n n matrix of rank n 1 and with zero row sums and prove some properties known results for distance matrices of trees are derived as special cases in particular we obtain a formula due to graham and lovasz for the inverse of the distance matrix of a tree it is shown that if d is a nonsingular edm and l is the associated laplacian then d 1 l is nonsingular and has a nonnegative inverse finally infinitely divisible matrices are constructed using edms

finite graphs and amenability hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years recently unexpected applications of hyperfiniteness were found in computer science in the context of testability of graph properties in this paper we propose a unified approach to hyperfiniteness we establish some new results and give new proofs of theorems of schramm lovasz newman sohler and ornstein weiss

every minor closed property of sparse graphs is testable abstract suppose g is a graph of bounded degree d and one needs to remove n of its edges in order to make it planar we show that in this case the statistics of local neighborhoods around vertices of g is far from the statistics of local neighborhoods around vertices of any planar graph g in fact a similar result is proved for any minor closed property of bounded degree graphs the main motivation of the above result comes from theoretical computer science using our main result we infer that for any minor closed property p there is a constant time algorithm for detecting if a graph is far from satisfying p this in particular answers an open problem of goldreich and ron stoc 1997 20 who asked if such an algorithm exists when p is the graph property of being planar the proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs which rely on martingale arguments

product distance matrix of a tree with matrix weights abstract let t be a tree on n vertices and let the n 1 edges e 1 e 2 e n 1 have weights that are s s matrices w 1 w 2 w n 1 respectively for two vertices i j let the unique ordered path between i and j be p i j e r 1 e r 2 e r k define the distance between i and j as the s s matrix e i j p 1 k w e p consider the n s n s matrix d whose i j th block is the matrix e i j we give a formula for det d and for its inverse when it exists these generalize known results for the product distance matrix when the weights are real numbers

graphs whose distance matrix has at most three negative eigenvalues abstract let d g be the distance matrix of a connected simple graph g the negative inertia of d g denoted by n d g is the number of negative eigenvalues of d g in this paper we determine all connected graphs g whose distance matrix d g has at most three negative eigenvalues

determinant of the distance matrix of a tree with matrix weights let t be a tree with n vertices and let d be the distance matrix of t according to a classical result due to graham and pollack the determinant of d is a function of n but does not depend on t we allow the edges of t to carry weights which are square matrices of a fixed order the distance matrix d of t is then defined in a natural way we obtain a formula for the determinant of d which involves only the determinants of the sum and the product of the weight matrices

numerical analysis of low frequency electromagnetic scattering from three dimensional arbitrarily shaped conducting bodies using an inductance matrix we have proposed a numerical method for calculating low frequency electromagnetic scattering from three dimensional arbitrarily shaped perfectly conducting bodies using an inductance matrix the surface of the perfectly conducting scatterer and its apertures are modeled by a set of triangular patches the current on each patch is computed by solving a coupled inductance matrix equation scattering from various conducting bodies with apertures is calculated to compare with the other methods the proposed method well agrees with finite element method in electromagnetic field distribution

on the gap of finite metric spaces of p negative type let x d be a metric space of p negative type recently i doust and a weston introduced a quantification of the p negative type property the so called gap of x this paper gives some formulas for the gap of a finite metric space of strict p negative type and applies them to evaluate for some concrete finite metric spaces

singularities of rational functions and minimal factorizations the noncommutative and the commutative setting we show that the singularities of a matrix valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization the proof uses a new notion of noncommutative backward shifts as an application we establish the commutative counterpart of the singularities theorem the singularities of a matrix valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its fornasini marchesini realizations with the minimal possible state space dimension the singularities results imply the absence of zero pole cancellations in a minimal factorization both in the noncommutative and in the commutative setting

the inverse of the distance matrix of a distance well defined graph abstract a square matrix l is called a laplacian like matrix if l j 0 and j t l 0 a square matrix d is left or right laplacian expressible if there exist a number 0 a column vector satisfying t j 1 and a square matrix l such that t d j t l d i j t and l j 0 or d j d l i j t and j t l 0 we consider the generalized distance matrix d see definition 4 1 of a graph whose blocks correspond to left or right laplacian expressible matrices then d is also left or right laplacian expressible and the inverse d 1 when it exists can be expressed as the sum of a laplacian like matrix and a rank one matrix

computing the inversion height of some embeddings of the free algebra and the free group algebra abstract let x be a set and k a field we show the so called jf embeddings of k x in a division ring have inversion height at most two and give examples of inversion height one and two we use these embeddings to obtain embeddings of the free group algebra in a division ring of inversion height one and two

someinverseproblemsforellipticmatriceswith zero diagonal abstract we present explicit formulae which allow us to construct elliptic matrices with zero diagonal from a given spectrum as well as from the complete interlacing system

a note on the relationship between the conjugate gradient method and polynomials orthogonal over the spectrum of a linear operator the relationship between the conjugate gradient method and polynomials orthogonal over the spectrum of a linear operator is discussed it is shown that as a byproduct of the conjugate gradient construction two sets of polynomials are generated which are orthogonal with respect to a positive real measure over the spectrum of a self adjoint operator the roots of these polynomials can under certain circumstances be used to track the eigenvalues or singular values of the relevant operator

l2 spectral invariants and convergent sequences of finite graphs abstract using the spectral theory of weakly convergent sequences of finite graphs we prove the uniform existence of the integrated density of states for a large class of infinite graphs

theoretical aspects of the physical realizability of broad band equivalent circuits for energy collecting structures the practical implementation of equivalent circuits that represent the energy delivered to an arbitrary load impedance by an antenna or a port on a scatterer is discussed particular attention is paid to the positive real pr function issues associated with the equivalent admittance of the structure when evolved from the point of view of the singularity expansion method sem a proof is given which establishes that the eigenadmittances reciprocal eigenvalues of a structure are positive real functions the connection between this fact and the positive realness of the admittance associated with individual current eigenmode contributions to total current is discussed an ad hoc approach to approximating these admittances with simpler pole pair positive real admittances is given for high q structures a brief survey of the applicable circuit synthesis algorithms and their features is provided some fundamental considerations in the implementation of the sources representing the energy coupled from the incident wave are presented

loop star basis functions and a robust preconditioner for efie scattering problems an electric field integral equation efie formulation using the loop star basis functions has been developed for modeling plane wave scattering from perfect conducting objects a stability analysis at the dc limit shows that the use of the rao wilton glisson rwg basis functions results in a singular matrix operator however the use of the loop star basis functions results in a well conditioned matrix moreover a preconditioner constructed from a two step process based on near interactions and an incomplete factorization with a heuristic drop strategy has been proposed in conjunction with the conjugate gradient method to solve the resulting matrix equation the approach is shown to be effective for resolving both the low frequency instability and the bad conditioning of the efie method the computational complexity of the proposed approach is shown to be o n sup 2

every property of hyperfinite graphs is testable a k disc around a vertex v of a graph g v e is the subgraph induced by all vertices of distance at most k from v we show that the structure of a planar graph on n vertices and with constant maximum degree d is determined up to the modification insertion or deletion of at most epsilon d n edges by the frequency of k discs for certain k k epsilon d that is independent of the size of the graph we can replace planar graphs by any hyperfinite class of graphs which includes for example every graph class that does not contain a set of forbidden minors a pure combinatorial consequence of this result is that two d bounded degree graphs that have similar frequency vectors that is the ell 1 difference between the frequency vectors is small are close to isomorphic where close here means that by inserting or deleting not too many edges in one of them it becomes isomorphic to the other we also obtain the following new results in the area of property testing which are es

loop star and loop tree decompositions analysis and efficient algorithms a new analysis of the spectral properties of loop star and loop tree decompositions is presented in this work the analysis will shed light on the behavior of these decompositions when used with regular operators such as the magnetic field and the calderon preconditioned electric field integral operators this work will explain the ill conditioning problems reported in literature and will provide a family of efficient algorithms to solve the ill conditioning and regularizing several loop star tree decomposed equations of interest for applications the theory will be corroborated by numerical results that will show the practical impact of the theoretical developments

an accurate broad band method of moments using higher order basis functions and tree loop decomposition a method is presented that combines the low frequency accuracy of tree loop decomposition with higher order basis functions to achieve an electric field integral equation based method of moments that is accurate over a wide band of frequencies in general the use of higher order divergence conforming bases introduces more solenoidal degrees of freedom than the standard rao wilton glisson zeroth order bases these solenoidal degrees of freedom are counted and identified and an efficient method to construct the required higher order loops is presented numerical results demonstrate that the technique allows the use of a single mesh to acquire scattering results over a wide band of frequencies using arbitrary order basis functions

distance energy change of complete bipartite graph due to edge deletion abstract the distance matrix distance eigenvalue and distance energy of a connected graph have been studied intensively in the literature we propose a new problem of studying how the distance energy changes when an edge is deleted in this paper we prove that the distance energy of a complete bipartite graph is always increased when an edge is deleted

the coefficients of the immanantal polynomial abstract an expression of the coefficient of immanantal polynomial of an n n matrix is present moreover we give expressions of the coefficient of immanantal polynomials of combinatorial matrices adjacency matrix laplacian matrix signless laplacian matrix as applications we show that the immanantal polynomials for laplacian matrix and signless laplacian matrix of bipartite graphs are the same this is a generalization of the characteristic polynomial for laplacian matrix and signless laplacian matrix of bipartite graphs furthermore we consider the relations between the characteristic polynomial and the immanantal polynomial for trees

uniformly recurrent subgroups and simple c algebras we study uniformly recurrent subgroups urs introduced by glasner and weiss 18 answering their query we show that any urs z of a finitely generated group is the stability system of a minimal z proper action we also show that for any sofic urs z there is a z proper action admitting an invariant measure we prove that for a urs z all z proper actions admits an invariant measure if and only if z is coamenable in the second part of the paper we study the separable c algebras associated to urs s we prove that if a urs is generic then its c algebra is simple we give various examples of generic urs s with exact and nuclear c algebras and an example of a urs z for which the associated simple c algebra is not exact and not even locally reflexive in particular it admits both a uniformly amenable trace and a nonuniformly amenable trace

well conditioned muller formulation for electromagnetic scattering by dielectric objects numerical solution of electromagnetic scattering by homogeneous dielectric objects with the method of moments mom and rao wilton glisson rwg basis functions is discussed it is shown that the low frequency breakdown associated to the mom solution of scattering by dielectric objects can be avoided by the classical muller formulation without the loop tree or loop star basis functions two variations of the muller formulation t mu spl uml ller and n muller are considered it is demonstrated that only the n muller formulation with the galerkin method and rwg functions gives stable solution discretization of the n muller formulation leads to a well conditioned matrix equation and rapidly converging iterative solutions on a wide frequency range from very low frequencies to microwave frequencies at zero frequency the n muller formulation decouples into the electrostatic and magnetostatic integral equations

infinite dimensional representations of finite dimensional algebras and amenability we present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits we introduce the rank spectrum of a finite dimensional algebra r over a countable field the elements of the rank spectrum are representations of the algebra into von neumann regular rank algebras and two representations are considered to be equivalent if they induce the same sylvester rank functions on r matrices based on this approach we can divide the finite dimensional algebras into three types finite amenable and non amenable representation types we prove that string algebras are of amenable representation type but the wild kronecker algebras are not here the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part we also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by john von neumann in the 1930 s as an application in algorithm theory we introduce and study the notion of parameter testing of modules over finite dimensional algebras that is analogous to the testing of bounded degree graphs introduced by goldreich and ron we shall see that for string algebras all the reasonable stable parameters are testable

laplacian immanantal polynomials and the gts poset on trees abstract let t be a tree on n vertices with laplacian l t and let gts n be the generalised tree shift poset on the set of unlabelled trees on n vertices inequalities are known for coefficients of the characteristic polynomial of l t as we go up the poset gts n in this work we generalise these inequalities to the q laplacian l t q of t and to the coefficients of all immanantal polynomials

eigenvalue monotonicity of q laplacians of trees along a poset abstract let t be a tree on n vertices with q laplacian matrix l t q let gts n be the generalized tree shift poset on the set of unlabelled trees with n vertices we prove that for all q r going up on gts n has the following effect the spectral radius of l t q increases while the smallest eigenvalue of l t q decreases we also prove that for all q r with q 1 going up on gts n increases the second smallest eigenvalue of l t q these generalize known results for eigenvalues of the laplacian matrix of trees as a corollary we obtain consequences about the eigenvalues of q t laplacians and exponential distance matrices of trees

approximation algorithm for connected bounded degree deletion problem on unit disk graphs abstract in this paper we study the minimum connected k bounded degree node deletion problem min c kbdnd for a connected graph g a constant k and a weight function w v r a vertex set c v g is a kbdnd set if the maximum degree of graph g c is at most k if furthermore the subgraph of g induced by c is connected then c is a ckbdnd set the goal of minwkbdnd resp minwckbdnd is to find a kbdnd set resp ckbdnd set with the minimum weight in this paper we focus on their cardinality versions with w v 1 v v which are denoted as minkbdnd and minckbdnd this paper presents a 1 e and a 3 76 approximation algorithm for minkbdnd and minckbdnd on unit disk graphs respectively where 0 e 1 is an arbitrary constant

equivalent circuit synthesis for a loop antenna based on the singularity expansion method the singularity expansion method sem is used to derive active equivalent circuits for a plane wave excited loop antenna immersed in a lossless and homogeneous medium the circuits consist of simple modules associated with the dominant pole pairs of the loop and circuit element values are expressed in terms of the sem data particular attention is paid to the issue of physical realizability the synthesized circuits have been analyzed under various excitation and loading conditions by a general purpose circuit analysis code and the results agree favorably with the responses of the loop computed by other means

inverse of the distance matrix of a cycle clique graph abstract a connected graph g all of whose blocks are cycles or cliques is called a cycle clique graph let d be the distance matrix of g by a theorem of graham et al we have det d 0 if all cycle blocks have odd vertices in this paper we give the formula for the inverse of d

the inertia and energy of the distance matrix of a connected graph abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we show that the distance matrix of a clique tree is non singular moreover we also prove that the distance matrix of a clique tree has exactly one positive d eigenvalue in addition we determine the extremal graphs with maximum and minimum distance energy among all clique trees

on the distance spectral radius of bipartite graphs abstract in this paper we determine the unique graph with minimum distance spectral radius among all connected bipartite graphs of order n with a given matching number moreover we characterize the graphs with minimal distance spectral radius in the class of all connected bipartite graphs with a given vertex connectivity

a q analogue of the distance matrix of a tree we consider a q analogue of the distance matrix called the q distance matrix of an unweighted tree and give formulae for the inverse and the determinant which generalize the existing formulae for the distance matrix we obtain the smith normal form of the q distance matrix of a tree the relationship of the q distance matrix with the laplacian matrix leads to q analogue of the laplacian matrix of a tree some of whose properties are also given we study another matrix related to the distance matrix the exponential distance matrix and show its relationship with the q laplacian and the q distance matrix a formula for the determinant of the q distance matrix of a weighted tree is also given

hook immanantal and hadamard inequalities for q laplacians of trees abstract let t be a tree on n vertices with laplacian matrix l and q laplacian l q let k be the character of the irreducible representation of s n indexed by the hook partition k 1 n k and let d k l be the normalized hook immanant of l corresponding to the character k inequalities for d k l as k increases are known by assigning a statistic to vertex orientations on trees we generalize these inequalities to the q analogue l q of l for all q r and to the bivariate q t laplacian l q t for some values q t our statistic based approach also generalizes several other inequalities including the changing index k l of the hadamard inequality for l to the matrix l q and l q t thus we extend several results about l to l q which includes the case when l q is not positive semidefinite

kesten s theorem for invariant random subgroups an invariant random subgroup of the countable group is a ran dom subgroup of whose distribution is invariant under conjuga tion by all elements of we prove that for a nonamenable invariant random subgroup h the spectral radius of every finitely supported random walk on i s strictly less than the spectral radius of the corresponding random walk on h this generalizes a result of kesten who proved this for normal subgroups as a byproduct we show that for a cayley graph g of a linear group with no amenable normal subgroups any sequence of finite quotients of g that spectrally approximates g converges to g in benjamini schramm convergence in particular this implies that infinite sequences of finite d regular ramanujan schreier graphs have essentially large girth

characterization of symmetric m matrices as resistive inverses abstract we aim here at characterizing those nonnegative matrices whose inverse is an irreducible stieltjes matrix specifically we prove that any irreducible stieltjes matrix is a resistive inverse to do this we consider the network defined by the off diagonal entries of the matrix and we identify the matrix with a positive definite schrodinger operator whose ground state is determined by the lowest eigenvalue of the matrix and the corresponding positive eigenvector we also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric m matrices

on the existence of branch points in the eigenvalues of the electric field integral equation operator in the complex frequency plane it is demonstrated by specific examples that eigenvalues of the electric field integral equation efie operator for finite extent objects in lossless media can have branch points in the complex frequency s plane specifically it is shown from the analysis of the spheroidal wave equation in the s plane that branch points are present in the eigenvalues of the integral operator associated with the scalar scattering problem from a perfectly conducting spheroid similarly it is concluded from the analysis of the mathieu s differential equation that branch points besides that associated with the infinite extent of the object exist in the s plane behavior of the eigenvalues of the efie operator for a perfectly conducting infinite elliptic cylinder a proof is given that branch point singularities cannot occur in the eigenvalues of efie operators for structures in which geometrical symmetry completely determines the eigenfunctions and hence they are frequency independent it is conjectured that branch points may always be present when sufficient object symmetry is lacking this conjecture is supported by the fact that branch points appear when a sphere is deformed into a spheroid or when a circular cylinder is deformed into an elliptic one an analogous phenomenon has been observed in circuit and transmission line problems for example it is shown that when the taper parameter of an exponential transmission line goes to zero the line becomes uniform and thus a symmetrical structure the branch points in the eigenvalues of the impedance matrix move away to infinity

borel chromatic numbers we study in this paper graph coloring problems in the context of descriptive set theory we consider graphs g x r where the vertex set x is a standard borel space i e a complete separable metrizable space equipped with its algebra of borel sets and the edge relation r x 2 is definable i e borel analytic co analytic etc a borel n coloring of such a graph where 1 n n 0 is a borel map c x y with card y n such that xry c x c y if such a borel coloring exists we define the borel chromatic number of g in symbols x b g to be the smallest such n otherwise we say that g has uncountable borel chromatic number in symbols x b g n 0