the determinant of a tree s neighborhood matrix abstract let n be an n n neighborhood matrix for a tree or forest we show that det n is bounded by the nth fibonacci number we obtain a simple elegant algorithm to compute det n that operates directly on the forest and uses o n space and o n arithmetic operations

connected graphs of fixed order and size with maximal index some spectral bounds the index or spectral radius of a simple graph is the largest eigenvalue of its adjacency matrix for connected graphs of fixed order and size the graphs with maximal index are not yet identified in the general case it is known for a long time that these graphs are nested split graphs or threshold graphs in this paper we use the eigenvector techniques for getting some new lower and upper bounds on the index of nested split graphs besides we give some computational results in order to compare these bounds

spectral ordering of trees with small index abstract studying graphs by means of the largest eigenvalue of the adjacency matrix the index has been a problem largely investigated in spectral graph theory here we consider trees with index in the real interval 2 2 5 and their ordering with respect to the index for such graphs we prove three indices comparisons whose order has been conjectured in belardo et al 2008 2 by doing so we prove a conjectured ordering of trees with smallest indices open since 2002 and we obtain the first 8 trees of even order with largest index

algebraic graph theory without orientation abstract let g be an undirected graph with vertices v 1 v 2 v and edges e1 e2 e let m be the matrix whose ijth entry is 1 if ej is a link incident with vi 2 if ej is a loop at vi and 0 otherwise the matrix obtained by orienting the edges of a loopless graph g i e changing one of the 1 s to a 1 in each column of m has been studied extensively in the literature the purpose of this paper is to explore the substructures of g and the vector spaces associated with the matrix m without imposing such an orientation we describe explicitly bases for the kernel and range of the linear transformation from r to r defined by m our main results are determinantal formulas using the unoriented laplacian matrix mmt to count certain spanning substructures of g these formulas may be viewed as generalizations of the matrix tree theorem the point of view adopted in this paper also gives rise to a matroid structure on the edges of g analogous to the cycle matroid and its dual in this setting the analogue of a spanning forest can have components with one odd cycle and the analogue of an edge cut has the property that its removal creates a new bipartite component

parameterizing cut sets in a graph by the number of their components for a connected graph g v e a subset u v is a disconnected cut if u disconnects g and the subgraph gu induced by u is disconnected as well a cut u is a k cut if gu contains exactly k 1 components more specifically a k cut u is a k cut if v u induces a subgraph with exactly 2 components the disconnected cut problem is to test whether a graph has a disconnected cut and is known to be np complete the problems k cut and k cut are to test whether a graph has a k cut or k cut respectively by pinpointing a close relationship to graph contractibility problems we show that k cut is in p for k 1 and any fixed constant 2 while it is np complete for any fixed pair k 2 we then prove that k cut is in p for k 1 and np complete for any fixed k 2 on the other hand for every fixed integer g 0 we present an fpt algorithm that solves k cut on graphs of euler genus at most g when parameterized by k by modifying this algorithm we can also show that k cut is in fpt for this graph class when parameterized by k finally we show that disconnected cut is solvable in polynomial time for minor closed classes of graphs excluding some apex graph

eigenvalue location for chain graphs abstract chain graphs also called double nested graphs play an important role in the spectral graph theory since every connected bipartite graph of fixed order and size with maximal largest eigenvalue is a chain graph in this paper for a given chain graph g we present an algorithmic procedure for obtaining a diagonal matrix congruent to a x i where a is the adjacency matrix of g and x any real number using this procedure we show that any chain graph has its least positive eigenvalue greater than 1 2 and also prove that this bound is best possible a similar procedure for threshold graphs also called nested split graphs is outlined

trees with 4 or 5 distinct normalized laplacian eigenvalues abstract we develop a tool for locating eigenvalues of the normalized laplacian matrix of trees this is obtained by extending an algorithm designed for the adjacency matrix due to jacobs and trevisan 2011 as an application we study the multiplicity of normalized laplacian eigenvalues of small diameter trees our main result is the characterization of the trees that have 4 or 5 distinct normalized laplacian eigenvalues we also show that with a fixed diameter these trees are determined by their normalized laplacian spectrum

an explicit formula for eigenvalues of bethe trees and upper bounds on the largest eigenvalue of any tree a bethe tree bd k is a rooted unweighted of k levels in which the root vertex has degree equal to d the vertices at level j 2 j k 1 have degree equal to d 1 and the vertices at level k are the pendant vertices in this paper we first derive an explicit formula for the eigenvalues of the adjacency matrix of bd k moreover we give the corresponding multiplicities next we derive an explicit formula for the simple nonzero eigenvalues among them the largest eigenvalue of the laplacian matrix of bd k finally we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the laplacian matrix of any tree t these upper bounds are given in terms of the largest vertex degree and the radius of t and they are attained if and only if t is a bethe tree

applications of recurrence relations for the characteristic polynomials of bethe trees a bethe tree of k levels b k d is a rooted tree such that the root vertex has degree d the vertices from level 2 to k 1 have degree d 1 and the vertices at level k are leaves in this paper we obtain a recurrence relation for the characteristic polynomial and the laplacian characteristic polynomial of bethe trees as an application we prove that there are no integral bethe trees except for the star b 2 n 2 and we obtain a recurrence relation for the laplacian energy like invariant of bethe trees

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

hitting minors on bounded treewidth graphs ii single exponential algorithms abstract for a finite collection of graphs f the f m deletion resp f tm deletion problem consists in given a graph g and an integer k decide whether there exists s v g with s k such that g s does not contain any of the graphs in f as a minor resp topological minor we are interested in the parameterized complexity of both problems when the parameter is the treewidth of g denoted by tw and specifically in the cases where f contains a single connected planar graph h we present algorithms running in time 2 o tw n o 1 called single exponential when h is either p 3 p 4 c 4 the paw the chair and the banner for both h m deletion and h tm deletion and when h k 1 i with i 1 for h tm deletion some of these algorithms use the rank based approach introduced by bodlaender et al 2015 7 this is the second of a series of articles on this topic and the results given here together with other ones allow us in particular to provide a tight dichotomy on the complexity of h m deletion in terms of h

irreducible decomposition of powers of edge ideals abstract in this paper by using some tools from graph theory mainly the theorem on ear decomposition of factor critical graphs given by l lovasz and the canonical decomposition of a graph given by edmonds and gallai we can describe each irreducible component of powers of edge ideals of a graph as an application we use our results to persistence of irreducible components to bound dstab and to count the number of irreducible components of powers of edge ideals of a graph

compatibility of unrooted phylogenetic trees is fpt a collection of t1 t2 tk of unrooted leaf labelled phylogenetic trees all with different leaf sets is said to be compatible if there exists a tree t such that each tree ti can be obtained from t by deleting leaves and contracting edges determining compatibility is np hard and the fastest algorithm to date has worst case complexity of around nk time n being the number of leaves here we present an o nf k algorithm proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable fpt with respect to the number k of trees

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

on the sum of the laplacian eigenvalues of a tree given an n vertex graph g v e the laplacian spectrum of g is the set of eigenvalues of the laplacian matrix l d a where d and a denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively in this paper we study the laplacian spectrum of trees more precisely we find a new upper bound on the sum of the k largest laplacian eigenvalues of every n vertex tree where k 1 n this result is used to establish that the n vertex star has the highest laplacian energy over all n vertex trees which answers affirmatively to a question raised by radenkovic and gutman 10

on the distance spectra of threshold graphs abstract a graph is called a threshold graph if it does not contain induced c 4 p 4 or 2 k 2 such graphs have numerous applications in computer science and psychology and they also have nice spectral properties in this paper we consider the distance matrix of a connected threshold graph we show that there are no distance eigenvalues of threshold graphs lying in the interval 2 1 and all the eigenvalues other than 2 or 1 are simple besides we determine all threshold graphs with distinct distance eigenvalues

integer lu factorizations abstract let a be an m by n integer matrix and r rank a a necessary and sufficient condition is given for a to have an integer lu factorization and a modification of gaussian elimination is given for finding such factorizations when the first r leading principal minors are nonzero an algorithm is given for ordering the vertices of a tree with a loop at its root so that its adjacency matrix has an integer lu factorization

eigenvalue free interval for threshold graphs abstract this paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which 1 and 0 are considered as trivial eigenvalues we show that threshold graphs have no non trivial eigenvalues in the interval 1 2 2 1 2 2 this confirms a conjecture by aguilar lee piato and schweitzer 2018

locating the eigenvalues of trees abstract e give an o n method that computes for any tree t and interval how many eigenvalues of t lie within the interval our method is based on sylvester s law of inertia we use our algorithm to show that the nonzero eigenvalues of a caterpillar are simple it follows that caterpillars having b back nodes where b 2 1 are not integral we also show that among the regular caterpillars c b k formed by adjoining k legs to each of b back nodes all positive roots are in the interval k 1 k 2 and c b k is not integral if b 2

equivalences and transformations of regular systems applications to recursive program schemes and grammars abstract this work presents a unified theory of recursive program schemes context free grammars grammars on arbitrary algebraic structures and in fact recursive definitions of all kind by means of regular systems the equivalences of regular systems associated with either all their solutions or their least solutions in all domains of appropriate type satisfying a set of algebraic laws expressed by equations are systematically investigated and characterized in some cases in terms of system transformations by folding unfolding and rewriting according to the equational algebraic laws grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union and the same questions are raised for them we also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system this theory applies to grammars of many kinds which generate trees graphs etc we formulate some classical transformations of context free grammars in terms of correct transformations which only use folding unfolding and algebraic laws and we immediately obtain their correctness

combinatorial characterization of the null spaces of symmetric h matrices abstract we characterize the structure of null spaces of symmetric diagonally dominant sdd matrices and symmetric h matrices with non negative diagonal entries we show that the structure of the null space of a symmetric sdd matrix or h matrix a depends on the structure of the connected components of its underlying graph each connected component contributes at most one vector to the null space this paper provides a combinatorial characterization of the rank of each connected component and a combinatorial characterization of a null vector if one exists for sdd matrices we also present an efficient combinatorial algorithm for constructing an orthonormal basis for the null space the paper also shows a close connection between gain graphs and h matrices which extends known results regarding the connection between undirected graphs and laplacian matrices and between signed graphs and sdd matrices we show how to exploit these combinatorial algorithms to reliably solve certain singular linear systems in finite precision arithmetic

on the normalized spectrum of threshold graphs abstract in this article we investigate normalized adjacency eigenvalues simply normalized eigenvalues and normalized adjacency energy of connected threshold graphs a threshold graph can always be represented as a unique binary string certain eigenvalues are obtained directly from its binary representation and the rest of the eigenvalues are evaluated from its normalized equitable partition matrix finally we characterize threshold graphs with at most five distinct eigenvalues

the determinant of a unicyclic graph s neighborhood matrix abstract let g be a unicyclic graph with n vertices and the unique cycle c a g and n g its adjacency matrix and neighborhood matrix respectively and a scalar we obtain an algorithm for computing the determinant of in a g which uses o n space and o n arithmetic operations for n where min dg x x v c applications include computing the determinants of a g and n g and computing the characteristic polynomial of a g

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

on the distribution of eigenvalues of a simple undirected graph abstract for a simple undirected graph g n let i g n be the ith largest eigenvalue of g n this paper presents mainly the following 1 for n 4 if g n is incomplete then n 2 n g n 1 1 4 n 3 n 1 2 2 seven sufficient and necessary conditions such that 2 g n 1 3 3 g n 1 implies that j g n 1 j 3 4 n 1

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

computational complexity of compaction to reflexive cycles in this paper we solve a widely publicized open problem posed by peter winkler in 1988 the problem is to decide whether or not it is possible to partition the vertices of a graph into four distinct nonempty sets a b c and d such that there is no edge between the sets a and c and between the sets b and d and that there is at least one edge between any other pair of distinct sets winkler asked whether this problem is np complete we show in this paper that it is np complete we study the problem as the compaction problem for a reflexive 4 cycle we also show in this paper that the compaction problem for a reflexive k cycle is np complete for all k geq 4

indices for special classes of trees abstract in this article we study the index the largest eigenvalue of the adjacency matrix in two special classes of trees namely starlike trees and double brooms for each class we determine conditions for the index to be integer we examine conditions in order to compare the index of a starlike tree with 1 where is the maximum degree of the graph we also prove that indices of a broom like tree and a double broom except for some cases are between and 1 characterizing when occurs one equality furthermore we build in these classes infinite families of non integral trees with integer index

on nested split graphs whose second largest eigenvalue is less than 1 abstract we determine all nested split graphs i e graphs having no induced subgraphs equal to 2 k 2 p 4 or c 4 having the second largest eigenvalue less than 1 and give some data and observations regarding obtained results

an axiomatic definition of context free rewriting and its application to nlc graph grammars abstract an abstract notion of context free grammar is introduced it deals with abstract objects that can be words trees graphs or other combinatorial objects it is applied to nlc graph grammars introduced by rozenberg and janssens the monadic second order theory of a context free nlc set of graphs is decidable

model reduction via schur decomposition a simple procedure for model reduction based upon a schur decomposition of a given system is presented the obtained partitioned system allows the combination of singular perturbation and aggregation approaches

unimodular equivalence of graphs abstract let g be a simple graph with n vertices and m edges denote by q the vertex edge incidence matrix corresponding to some orientation of the edges of g define l g qqt and k g qtq then l g is the so called laplacian matrix and k g its edge version viewed as integer matrices the smith normal form of l g is complicated but the smith normal form of k g for connected graphs is always i n 2 n 0 m n 1 an edge version of the matrix tree theorem is used in the proof there is an application to digraph flows with constant resultants over abelian groups finally if l g and l h are congruent then g and h have the same chromatic polynomial

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

some new considerations about double nested graphs abstract in the set of all connected graphs with fixed order and size the graphs with maximal index are nested split graphs also called threshold graphs it was recently and independently observed in bell et al 2008 3 and bhattacharya et al 2008 4 that double nested graphs also called bipartite chain graphs play the same role within class of bipartite graphs in this paper we study some structural and spectral features of double nested graphs in studying the spectrum of double nested graphs we rather consider some weighted nonnegative matrices of significantly less order which preserve all positive eigenvalues of former ones moreover their inverse matrices appear to be tridiagonal using this fact we provide several new bounds on the index largest eigenvalue of double nested graphs and also deduce some bounds on eigenvector components for the index we conclude the paper by examining the questions related to main versus non main eigenvalues

graphs for which the least eigenvalue is minimal ii let g be a connected graph whose least eigenvalue g is minimal among the connected graphs of prescribed order and size we show first that either g is complete or g is a simple eigenvalue in the latter case the sign pattern of a corresponding eigenvector determines a partition of the vertex set and we study the structure of g in terms of this partition we find that g is either bipartite or the join of two graphs of a simple form

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

on the minors of an incidence matrix and its smith normal form abstract consider the vertex edge incidence matrix of an arbitrary undirected loopless graph we completely determine the possible minors of such a matrix these depend on the maximum number of vertex disjoint odd cycles i e the odd tulgeity of the graph the problem of determining this number is shown to be np hard turning to maximal minors we determine the rank of the incidence matrix this depends on the number of components of the graph containing no odd cycle we then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix as well as its smith normal form these results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs finally we give a sufficient condition for a system of equations whose coefficient matrix is an incidence matrix to admit an integer solution

optimal low order controller design via lqg like parametrization abstract a new lqg like parametrization scheme is introduced for low order controller design descent based design methodologies are developed for optimizing an h 2 cost with respect to the controller design parameters in this case the reduced order regulator and estimator gain matrices in a modified version the reduced order model of the plant is also updated numerical experiments using large space structure models show the efficacy of this approach

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

eigenvalue location in threshold graphs abstract let g be a threshold graph of order n with adjacency matrix a we present an o n algorithm for constructing a diagonal matrix congruent to b x a x i for any x an application using sylvester s law of inertia can determine how many eigenvalues lie in an interval allowing efficient approximation we prove that eigenvalues of threshold graphs other than 0 or 1 are simple we give the spectrum for threshold graphs g k j represented by 0 k 1 j when n 3 g n n 3 n 3 has the minimum eigenvalue min n among threshold graphs of order n and a formula for min n is given there is one more graph if n 2 mod 3

eigenvalue location in graphs of small clique width abstract finding a diagonal matrix congruent to a c i for constants c where a is the adjacency matrix of a graph g allows us to quickly tell the number of eigenvalues in a given interval if g has clique width k and a corresponding k expression is known then diagonalization can be done in time o poly k n where n is the order of g

characterizing trees with large laplacian energy we investigate the problem of ordering trees according to their laplacian energy more precisely given a positive integer n we find a class of cardinality approximately p n whose elements are the n vertex trees with largest laplacian energy the main tool for establishing this result is a new upper bound on the sum sk t of the k largest laplacian eigenvalues of an n vertex tree t with diameter at least four where k 2 f1 ng

the distribution of eigenvalues of graphs abstract we show that every limit point of the kth largest eigenvalues of graphs is a limit point of the k 1 th largest eigenvalues and we find out the smallest limit point of the kth largest eigenvalues and an upper bound of the limit points of the kth smallest eigenvalues for k 4 we prove that there exists a gap beyond the smallest limit point in which no point is the limit point of the kth largest eigenvalues for the third largest eigenvalues of a graph g with at least three vertices we obtain that 1 3 g 1 3 g 1 5 2 consequently if gc is not a complete bipartite graph plus isolated vertices 3 g 3 d n where d n is the complement of the double star s 1 n 3

the spectra of some trees and bounds for the largest eigenvalue of any tree abstract let t be an unweighted tree of k levels such that in each level the vertices have equal degree let nk j 1 and dk j 1 be the number of vertices and the degree of them in the level j we find the eigenvalues of the adjacency matrix and laplacian matrix of t for the case of two vertices in level 1 nk 2 including results concerning to their multiplicity they are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for these matrices are d j 1 2 j k while the diagonal entries are 0 0 1 in the case of the adjacency matrix and d1 d2 dk 1 dk 1 in the case of the laplacian matrix finally we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the laplacian matrix of any given tree

on the parameterized complexity of recognizing well covered r l graph an r ell partition of a graph g is a partition of its vertex set into r independent sets and ell cliques a graph is r ell if it admits an r ell partition a graph is well covered if every maximal independent set is also maximum a graph is r ell well covered if it is both r ell and well covered in this paper we consider two different decision problems in the r ell well covered graph problem r ell wcg for short we are given a graph g and the question is whether g is an r ell well covered graph in the well covered r ell graph problem wc r ell g for short we are given an r ell graph g together with an r ell partition of v g into r independent sets and ell cliques and the question is whether g is well covered we classify most of these problems into p conp complete np complete np hard or conp hard only the cases wc r 0 g for r ge 3 remain open in addition we consider the parameterized complexity of these problems for several choices of parameters such as the size alpha of a maximum independent set of the input graph its neighborhood diversity or the number ell of cliques in an r ell partition in particular we show that the parameterized problem of deciding whether a general graph is well covered parameterized by alpha can be reduced to the wc 0 ell g problem parameterized by ell and we prove that this latter problem is in xp but does not admit polynomial kernels unless mathsf conp subseteq mathsf np mathsf poly

no threshold graphs are cospectral abstract a threshold graph g on n vertices is defined by binary sequence of length n in this paper we present an explicit formula for computing the characteristic polynomial of a threshold graph from its binary sequence applications include obtaining a formula for the determinant of adjacency matrix of a threshold graph and showing that no two nonisomorphic threshold graphs are cospectral

eigenvalues and energy in threshold graphs abstract assuming a uniform random model of selecting creation sequences we show that almost every connected threshold graph has more negative than positive eigenvalues we show that no threshold graphs have eigenvalues in 1 0 sequences of equienergetic graphs are given including the striking result that for all n 3 there exist n 1 threshold graphs of order n 2 pairwise noncospectral each equienergetic to k n 2 for all n 8 we find an n vertex hyperenergetic threshold graph

efficient computation of the characteristic polynomial of a threshold graph an efficient algorithm is presented to compute the characteristic polynomial of a threshold graph threshold graphs were introduced by chvatal and hammer as well as by henderson and zalcstein in 1977 a threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex which is a vertex adjacent to all the other vertices threshold graphs are special kinds of cographs which themselves are special kinds of graphs of clique width 2 we obtain a running time of o n log 2 n for computing the characteristic polynomial while the previously fastest algorithm ran in quadratic time we improve the running time drastically in the case where there is a small number of alternations between 0 s and 1 s in the sequence defining a threshold graph a simple recurrence equation for the determinant of a weighted threshold graph matrix is presented the characteristic polynomial is computed in o n log 2 n arithmetic operations the algorithm is more efficient if the alternations in the defining sequence are o n as the numbers may be of length n log n the bit complexity is investigated too

linear time lup decomposition of forest like matrices a square matrix m has an lup decomposition if it can be written as lup where l is a unit lower triangular matrix u is an upper triangular matrix and p is a permutation matrix we present a linear time algorithm for finding an lup decomposition for a nonsingular neighborhood matrix of a tree we also identify a more general class of matrices we call forest like for which the algorithm can be modified