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

on the laplacian eigenvalues of a graph abstract in the note we present an upper bound for the spectral radius of laplacian matrix of a graph in terms of a 2 degree of a vertex

laplacian spectral characterization of 3 rose graphs abstract a 3 rose graph is a graph consisting of three cycles intersecting in a common vertex j wang et al showed all 3 rose graphs with at least one triangle are determined by their laplacian spectra in this paper we complete the above laplacian spectral characterization and prove that all 3 rose graphs are determined by their laplacian spectra

invariant subspaces and hyper reflexivity for free semigroup algebras in this paper we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras this enables us to prove that they are reflexive and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces such algebras are called hyper reflexive this property is very strong but it has been established in only a very few cases moreover the prototypes of this class of algebras are the natural candidate for a non commutative analytic toeplitz algebra on n variables the case we make for this analogy is very compelling in particular in this paper the key to the invariant subspace analysis is a good analogue of the beurling theorem for invariant subspaces of the unilateral shift this leads to a notion of inner outer factorization in these algebras in a sequel to this paper 13 we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in cn a free semigroup algebra is the weak operator topology closed algebra generated by a set s1 sn of isometries with pairwise orthogonal ranges these conditions are described algebraically by s i sj iji for 1 i j n f

graph zn and some graphs related to zn are determined by their spectrum abstract it is proved that graph zn is determined by its adjacency spectrum as well as its laplacian spectrum z n 1 z n 2 z n k is determined by its adjacency spectrum where n1 n2 nk are integers at least 2 wn is not determined by its adjacency spectrum but is determined by its laplacian spectrum kzn tn are determined by their laplacian spectrum respectively where k is a positive integer

effect of weak prey in leslie gower predator prey model a leslie gower prey predator model incorporating partial prey protection is proposed prey densities are divided into two groups strong and weak the special dynamic of each group is considered the condition of existence and uniqueness of the equilibrium point in the positive octant is determined and by constructing a suitable lyapunov function it is shown that the unique equilibrium point is stable in the positive octant the effects of strong prey becoming weak prey is studied and it is shown that it has no influence on the persistent property of the system and can enhance the co existence of the prey predator

trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

laplacian graph eigenvectors abstract if g is a graph its laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix the main thrust of the present article is to prove several laplacian eigenvector principles which in certain cases can be used to deduce the effect on the spectrum of contracting adding or deleting edges and or of coalescing vertices one application is the construction of two isospectral graphs on 11 vertices having different degree sequences only one of which is bipartite and only one of which is decomposable

recursive boson system in the cuntz algebra o bosons and fermions are often written by elements of other algebras abe private communication gave a realization of bosons by formal infinite sums of the canonical generators of the cuntz algebra o we show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of o in this meaning we can regard as if the algebra b of bosons was a unital subalgebra of o on a given permutative representation according to this relation we compute branching laws arising from restrictions of representations of o on b for example it is shown that the fock representation of b is given as the restriction of the standard representation of o on b

the algebraic structure of non commutative analytic toeplitz algebras the non commutative analytic toeplitz algebra is the wot closed algebra generated by the left regular representation of the free semigroup on n generators we develop a detailed picture of the algebraic structure of this algebra in particular we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n ball the k dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of h over the unit disk in 6 17 18 20 a good case is made that the appropriate analogue for the analytic toeplitz algebra in n non commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators the papers cited obtain a compelling analogue of beurling s theorem and inner outer factorization in this paper we add further evidence the main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of cn the kernel is the subgroup of quasi inner automorphisms which are trivial modulo the wot closed commutator ideal additional evidence of analytic properties comes from the structure of k dimensional completely contractive representations which have a structure very similar to the fibration of the maximal ideal space of h over the unit disk an important tool in our analysis is a detailed structure theory for wot closed right ideals curiously left ideals remain more obscure the non commutative analytic toeplitz algebra ln is determined by the left regular representation of the free semigroup fn on n generators z1 zn which acts on 2 fn by w v wv for v w in fn in particular the algebra ln is the unital wot closed algebra generated by the isometries li zi for 1 i n this algebra and its norm closed version the noncommutative disk algebra were introduced by popescu 19 in an abstract sense in connection with a non commutative von neumann inequality and 1991 mathematics subject classification 47d25 march 9 1997 october 9 1997 final draft first author partially supported by an nserc grant and a killam research fellowship second author partially supported by an nsf grant

the ordering of trees and connected graphs by algebraic connectivity abstract in this paper we first determine that the first four trees of order n 9 with the smallest algebraic connectivity are p n q n w n and z n with p n q n w n z n t where t is any tree of order n other than p n q n w n and z n then we consider the effect on the laplacian eigenvalues of connected graphs by suitably adding edges by using these results and methods we finally determine that the first six connected graphs of order n 9 with the smallest algebraic connectivity are p n q n q n w n w n and w n with p n q n q n w n w n w n g where g is any connected graph of order n other than p n q n q n w n w n and w n

the graphs with spectral radius between 2 and 2 5 abstract we complete the determination of the graphs in the title begun by cvetkovic doob and gutman

starlike trees whose maximum degree exceed 4 are determined by their q spectra abstract let a g and d g be the adjacency matrix and the degree matrix of a graph g respectively the matrix d g a g is called the signless laplacian matrix of g the spectrum of the matrix d g a g is called the q spectrum of g a graph is said to be determined by its q spectrum if there is no other non isomorphic graph with the same q spectrum in this paper we prove that all starlike trees whose maximum degree exceed 4 are determined by their q spectra

laplacian integrality in p4 sparse and p4 extendible graphs abstract let g be a simple graph and l l g the laplacian matrix of g g is called l integral if all its laplacian eigenvalues are integer numbers it is known that every cograph a graph free of p4 is l integral the class of p4 sparse graphs and the class of p4 extendible graphs contain the cographs it seems natural to investigate if the graphs in these classes are still l integral in this paper we characterized the l integral graphs for both cases p4 sparse graphs and p4 extendible graphs

on some forests determined by their laplacian or signless laplacian spectrum we consider the class of graphs whose each component is either a proper subgraph of some smith graphs or belongs to a precized subset of smith graphs we classify the graphs from the considered class into those which are determined or not determined by laplacian or signless laplacian spectrum

atomic representations of cuntz algebras abstract to a representation of o n the cuntz algebra with n generators we associate a projection valued measure and we study the case when this measure has atoms the main technical tool are the spaces invariant for all the operators s i we classify the purely atomic representations and find when such representations are permutative applications include wavelet representations representations generated by finitely correlated states representation associated to hadamard triples or fractal spectral measures and representations associated to generalized walsh bases

the coefficients of laplacian characteristic polynomials of graphs abstract li et al j x li w c shiu a chang the number of spanning trees of a graph appl math lett 23 2010 286 290 obtained some upper bounds for the number of spanning trees of graphs in this paper we characterize the connected graph g with the connectivity resp edge connectivity and chromatic number which has the maximal coefficients of the laplacian characteristic polynomial since the number of spanning trees of a graph g is determined by one of the coefficients of the laplacian characteristic polynomial of g we generalize the results by li et al

stability analysis of a prey predator model with holling type iii response function incorporating a prey refuge in this paper we consider a prey predator model with holling type iii response function incorporating a prey refuge the purpose of the work is to offer mathematical analysis of the model and to discuss some significant qualitative results that are expected to arise from the interplay of biological forces some numerical simulations are carried out

the spectral characterization of graphs abstract a graph b r s is a graph consisting of two cycles c r 1 and c s 1 with just a vertex in common fan and luo in theorem 4 4 8 alleged that a graph is determined by its adjacency spectrum if it contains no cycle c 4 however according to theorem 6 3 16 we find that the result in theorem 4 4 8 is not completely correct in this paper we prove that b r s s r 7 is das if and only if s r 2 and b r s s r 7 has a unique cospectral mate if s r 2

the line graphs of lollipop graphs are determined by their spectra abstract haemers et al w h haemers x g liu y p zhang spectral characterizations of lollipop graphs linear algebra appl 428 2008 2415 2423 first investigated the spectral characterizations of the so called lollipop graphs h g l n with order n which is obtained by identifying a vertex of a cycle with order g and a pendent vertex of a path with order l for the graphs with least eigenvalue at least 2 cvetkovic and lepovic d cvetkovic m lepovic cospectral graphs with least eigenvalue at least 2 publ inst math nouv ser 78 92 2005 51 63 introduced the discriminant and the star value of a graph whose relations are investigated in this paper employing this relation and other techniques we prove that all line graphs of lollipop graphs are determined by their adjacency spectra

the complement of the path is determined by its spectrum abstract it is proved that a graph whose 0 1 adjacency matrix has the spectrum of p n the complement of the path on n vertices must be p n

graphs determined by polynomial invariants many polynomials have been defined associated to graphs like the characteristic matchings chromatic and tutte polynomials besides their intrinsic interest they encode useful combinatorial information about the given graph it is natural then to ask to what extent any of these polynomials determines a graph and in particular whether one can find graphs that can be uniquely determined by a given polynomial in this paper we survey known results in this area and at the same time we present some new results

branching laws for endomorphisms of fermions and the cuntz algebra o2 fermions are expressible in terms of polynomials in the canonical generators and their conjugates of the cuntz algebra o2 a canonical construction of such fermions is called a recursive fermion system in this paper various unital endomorphisms of the algebra of fermions are explicitly constructed by using the recursive fermion system we investigate how such endomorphisms transform the fock representation the infinite wedge representation and their duals

altering synchronizability by adding and deleting edges for scale free networks abstract in this paper we propose two methods for altering the synchronizability of scale free networks 1 adding edges between the max degree nodes and min degree nodes 2 deleting edges between the max degree nodes and max degree nodes after adding and deleting edges we find that the former adding process can weaken synchronizability while the latter deleting process can enhance it the two processes adding and deleting can preserve the scale free structure the study of the average clustering coefficient indicates that it is not the most closely correlated with the synchronizability among the topological features considered our work also suggests that there are some essential relations between the network synchronization and the dynamics of economic systems they can be used to deal with some problems in the real world such as relieving the economic crisis in addition the adding and deleting processes may have potential applications in modifying network structure in view of their low cost

degree maximal graphs are laplacian integral abstract an integer sequence d d 1 d 2 d n is graphic if there is a graph whose degree sequence is d a graph g is maximal if its degree sequence is majorized by no other graphic sequence the laplacian matrix of g is l g d g a g where d g is the diagonal matrix of vertex degrees and a g is the 0 1 adjacency matrix the article contains an explicit algorithmic construction of all maximal graphs from which it follows e g that apart from 0 the laplacian spectrum of a maximal graph is the conjugate of its degree sequence

on the aα characteristic polynomial of a graph abstract let g be a graph with n vertices and let a g and d g denote respectively the adjacency matrix and the degree matrix of g define a g d g 1 a g for any real 0 1 the a characteristic polynomial of g is defined to be det x i n a g j c j g x n j where det denotes the determinant of and i n is the identity matrix of size n the a spectrum of g consists of all roots of the a characteristic polynomial of g a graph g is said to be determined by its a spectrum if all graphs having the same a spectrum as g are isomorphic to g in this paper we first formulate the first four coefficients c 0 g c 1 g c 2 g and c 3 g of the a characteristic polynomial of g and then we observe that a spectra are much efficient for us to distinguish graphs by enumerating the a characteristic polynomials for all graphs on at most 10 vertices to verify this observation we characterize some graphs determined by their a spectra

graphs determined by their generalized characteristic polynomials abstract for a given graph g with 0 1 adjacency matrix a g the generalized characteristic polynomial of g is defined to be g g t det i a g td g where i is the identity matrix and d g is the diagonal degree matrix of g in this paper we are mainly concerned with the problem of characterizing a given graph g by its generalized characteristic polynomial g we show that graphs with the same generalized characteristic polynomials have the same degree sequence based on which a unified approach is proposed to show that some families of graphs are characterized by g we also provide a method for constructing graphs with the same generalized characteristic polynomial by using gm switching

extensions of representations of the car algebra to the cuntz algebra o2 the fock and the infinite wedge fermions are expressed by polynomials of canonical generators of the cuntz algebra o2 and they generate the u 1 fixed point subalgebra a o2u 1 of o2 by the canonical gauge action we extend the fock and the infinite wedge representations of a to permutative representations of o2 by these extensions the boson fermion correspondence is rewritten by canonical generators of o2

which q cospectral graphs have same degree sequences abstract let c n denote the cycle with n vertices and let g be the number of triangles of g if g is a graph with n vertices then 1 g 2 g n g denote the signless laplacian spectrum of g two graphs are said to be q cospectral if they have the same signless laplacian spectrum a graph g is said to be q d s if there is no other non isomorphic graph h such that g and h are q cospectral in this paper we prove that let g be a graph with n 12 vertices and 1 n g 2 g 5 n 1 g or let g be a graph with n 10 vertices and 0 n g 2 g 4 n 1 g if g and h are q cospectral then g and h share the same degree sequences and g h as a consequence of our results we show that all multi wheel graphs k 1 c q 1 c q 2 c q t are q d s

a note on the spectral characterization of θ graphs we consider graphs that is graphs obtained by subdividing the edges of the multigraph consisting of 3 parallel edges it is shown that any graph g is determined by the spectrum the multiset of eigenvalues except possibly when it contains a unique 4 cycle

domination number and laplacian eigenvalue of trees abstract a dominating set in a graph g is a vertex set s v g such that any vertex in v g s is adjacent to a vertex of s the domination number g is the minimum cardinality of a dominating set of g it is well known that g d g 1 3 where d g is the diameter of g let t be a tree on n vertices with diameter d t we show that t d t 1 3 if and only if t contains a laplacian eigenvalue of multiplicity n d t and such trees are completely determined as an application we show that some trees are determined by their laplacian spectra

on the ordering of trees by the laplacian coefficients abstract we generalize the results from x d zhang x p lv y h chen ordering trees by the laplacian coefficients linear algebra appl 2009 doi 10 1016 j laa 2009 04 018 on the partial ordering of trees with given diameter for two n vertex trees t 1 and t 2 if c k t 1 c k t 2 holds for all laplacian coefficients c k k 0 1 n we say that t 1 is dominated by t 2 and write t 1 c t 2 we proved that among n vertex trees with fixed diameter d the caterpillar c n d has minimal laplacian coefficients c k k 0 1 n the number of incomparable pairs of trees on 18 vertices is presented as well as infinite families of examples for two other partial orderings of trees recently proposed by mohar for every integer n we construct a chain t i i 0 m of n vertex trees of length n 2 4 such that t 0 s n t m p n and t i c t i 1 for all i 0 1 m 1 in addition the characterization of the partial ordering of starlike trees is established by the majorization inequalities of the pendent path lengths we determine the relations among the extremal trees with fixed maximum degree and with perfect matching and further support the laplacian coefficients as a measure of branching

unitary isomorphism of fock spaces of bosons and fermions arising from a representation of the cuntz algebra o2 bosons and fermions are described by using canonical generators of cuntz algebras on any permutative representation according to branching laws associated with these descriptions a certain representation of the cuntz algebra o2 induces fock representations hb and hf of bosons and fermions simultaneously from this a unitary operator u from hb to hf is obtained we show the explicit formula of the action of u on the standard basis of hb it is shown that u preserves the particle number of hb and hf

on determination of caterpillars with four terminal vertices by their laplacian spectrum abstract we consider the class of caterpillars with four terminal vertices here we prove that every of such caterpillar whose internal path differs in length from both 1 to 3 is uniquely determined by its laplacian spectrum next we take into consideration the remaining two possibilities for the internal path in the first situation we prove that there is exactly one caterpillar which is not determined by its laplacian spectrum while we find an infinite family of such caterpillars in the second finally some observations are given

on the laplacian coefficients of trees with a perfect matching let t be a tree with n vertices and let t k 0n 1 kck t n k be the characteristic polynomial of laplacian matrix of t it is well known that cn 2 t is equal to the wiener index of t while cn 3 t is equal to the modified hyper wiener index of t for two n vertex trees t1 and t2 write t1 t2 if ck t1 ck t2 for all 0 k n let n be the set of all trees with 2n vertices and perfect matchings except p2n and an 1 1 where ps is a path with s vertices and as t is a tree obtained from a star st 1 with t 1 vertices by attaching s pendent paths of length 2 to the center of st 1 in this paper we first give a graphic transformation that increases all laplacian coefficients of an arbitrary tree we then determine the minimum element and the second minimum element in n under the partial order and we finally identify the maximum element and second maximum element in n under the partial order furthermore we characterize some trees with extremal wiener indices and laplacian like energies

spectral characterizations of graphs with small spectral radius abstract a graph is said to have a small spectral radius if it does not exceed the corresponding hoffmann limit value in the case of signless laplacian matrix the hoffmann limit value is equal to 2 4 38 with being the real root of x 3 4 x 4 here the spectral characterization of connected graphs with small signless laplacian spectral radius is considered it is shown that all connected graphs with small laplacian spectral radius are determined by their laplacian spectra and all but one of connected graphs with small signless laplacian spectral radius are determined by their signless laplacian spectra

on bi regular graphs determined by their generalized characteristic polynomials abstract it was shown that whether a graph g is determined by its generalized characteristic polynomial is equivalent to the same problem for a well defined bi regular graph g also a unified approach is proposed to show that some new families of bi regular graphs are characterized by their generalized characteristic polynomials

on the signless laplacian spectral characterization of the line graphs of lollipop graphs abstract let h g k n denote the lollipop graph on n vertices obtained by identifying a vertex of the cycle c g of order g and an end vertex of the path p k 1 of order k 1 in this paper we prove that all line graphs l h g k n of lollipop graphs are determined by their laplacian spectrum furthermore we show that all l h g k n but the case that g k 3 are determined by their signless laplacian spectrum and also give the q cospectral class of l h r r n r 3

universal fermionization of bosons on permutative representations of the cuntz algebra o2 bosons and fermions are described by using canonical generators of cuntz algebras on any permutative representation we show a fermionization of bosons which universally holds on any permutative representation of the cuntz algebra o2 as examples we show fermionizations on the fock space and the infinite wedge

the algebraic connectivity of lollipop graphs abstract let c n g be the lollipop graph obtained by appending a g cycle c g to a pendant vertex of a path on n g vertices in 2002 fallat kirkland and pati proved that for n 3 g 1 2 and g 4 c n g c n g 1 in this paper we prove that for g 4 c n g c n g 1 for all n where c n g is the algebraic connectivity of c n g

note on the spectral characterization of some cubic graphs with maximum number of triangles abstract ubic graphs of given order with maximum number of triangles are characterized consequently it is proved that certain cubic graphs with maximum number of triangles are determined by their adjacency spectrum

signless laplacian spectral characterization of the cones over some regular graphs abstract let g be an r regular graph of order n we prove that the cone over g is determined by its signless laplacian spectrum for r 1 n 2 for r 2 and n 11 for r n 3 we show that the cone over g is determined by its signless laplacian spectrum if and only if the complement of g has no triangles a class of q cospectral graphs are also given

modelling and analysis of a harvested prey predator system incorporating a prey refuge the present paper deals with a prey predator model incorporating a prey refuge and independent harvesting in either species our study shows that using the harvesting efforts as controls it is possible to break the cyclic behaviour of the system and drive it to a required state the possibility of existence of bionomic equilibria has been considered the problem of optimal harvest policy is then solved by using pontryagin s maximal principle

qualitative analysis of a predator prey model with holling type ii functional response incorporating a prey refuge abstract we study a predator prey model with holling type ii functional response incorporating a prey refuge under homogeneous neumann boundary condition we show the existence and non existence of non constant positive steady state solutions depending on the constant m 0 1 which provides a condition for protecting 1 m u of prey u from predation moreover we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions

spectral bisection of graphs and connectedness abstract we present a refinement of the work of miroslav fiedler regarding bisections of irreducible matrices we consider graph bisections as defined by the cut set consisting of characteristic edges of the eigenvector corresponding to the smallest non zero eigenvalue of the graph laplacian the so called fiedler vector we provide a simple and transparent analysis including the cases when there exist components with value zero namely we extend the class of graphs for which the fiedler vector is guaranteed to produce connected subgraphs in the bisection furthermore we show that for all connected graphs there exist fiedler vectors such that connectedness is preserved by the bisection and estimate the measure of the set of connectedness preserving fiedler vectors with respect to the set of all fiedler vectors

on the spectral characterization of t shape trees a graph g is said to be determined by its spectrum if any graph having the same spectrum as g is isomorphic to g a t shape is a tree with exactly one of its vertices having maximal degree 3 in this paper we show that all t shape trees are determined by their spectra except for a few well defined cases

a note on sum of powers of the laplacian eigenvalues of bipartite graphs abstract for a graph g and a real number 0 the graph invariant s g is the sum of the th power of the non zero laplacian eigenvalues of g in this note we obtain some bounds of s g for a connected bipartite graph g which improve some known results of zhou b zhou on sum of powers of the laplacian eigenvalues of graphs linear algebra appl 429 2008 2239 2246

on algebraic connectivity and spectral integral variations of graphs abstract let g be a simple connected graph and l g be the laplacian matrix of g let a g be the second smallest eigenvalue of l g an eigenvector of l g corresponding to the eigenvalue a g is called a fiedler vector of g let y be a fiedler vector of g a characteristic vertex is a vertex u of g such that y u 0 and such that there is a vertex w adjacent to u satisfying y w 0 a characteristic edge is an edge u v such that y u y v

note the t shape tree is determined by its laplacian spectrum a t shape is a tree with exactly one of its vertices having maximal degree 3 it is proved in this paper that the t shape tree is determined by its laplacian spectrum

spectral determination of graphs whose components are paths and cycles we consider the class of graphs each of whose components is either a path or a cycle we classify the graphs from the class considered into those which are determined and those which are not determined by the adjacency spectrum in addition we compare the result with the corresponding results for the laplacian and the signless laplacian spectra it turns out that the signless laplacian spectrum performs the best confirming some expectations from the literature

graphs cospectral with starlike trees a tree which has exactly one vertex of degree greater than two is said to be starlike in spite of seemingly simple structure of these trees not much is known about their spectral properties in this paper we introduce a generalization of the notion of cospectrality called m cospectrality which turns out to be useful in constructing cospectral graphs based on this we construct cospectral mates for some starlike trees we also present a set of necessary and sufficient conditions for divisibility of the characteristic polynomial of a starlike tree by the characteristic polynomial of a path

laplacian spectrum characterization of extensions of vertices of wheel graphs and multi fan graphs the graph c n email protected k k is the product of a circuit c n 1 and a clique k k in this paper we will prove that it is determined by their laplacian spectrum except when n 1 6 if n 1 6 there are several counterexamples we also prove that the product of s vertex disjoint paths and a clique p n email protected p n email protected email protected p n s k k is also determined by the laplacian spectrum

on the laplacian coefficients of graphs under some transformations abstract we generalize three approaches on graph transformations respectively from stevanovic and ilic 2009 16 and tan 2011 19 we also generalize an approach of graph transformations on the spectral radius of adjacency matrix into the laplacian coefficients of graphs from li and feng 1979 26 moreover we determine the unique tree having the third maximal laplacian coefficients among all n vertex trees

starlike trees are determined by their laplacian spectrum in this paper we show that if g is a starlike tree then it is determined by its laplacian spectrum moreover we prove some facts about trees with the same adjacency spectrum as a starlike tree

on a delayed nonautonomous ratio dependent predator prey model with holling type functional response and diffusion abstract a two species periodic ratio dependent predator prey model with holling type functional response and diffusion is investigated three different sets of sufficient conditions one set is diffusive independent and the other two sets are diffusive dependent which ensure the permanence of the system and a set of sufficient condition which ensure the extinction of the predator species are obtained examples together with their numeric simulations show the feasibility of the main results

constructing graphs with given spectrum and the spectral radius at most 2 abstract two graphs are cospectral if their spectra coincide the set of all graphs that are cospectral to a given graph including the graph by itself is the cospectral equivalence class of the graph we say that a graph is determined by its spectrum or that it is a ds graph if it is a unique graph having that spectrum given n reals belonging to the interval 2 2 we want to find all graphs on n vertices having these reals as the eigenvalues of the adjacency matrix such graphs are called smith graphs our search is based on solving a system of linear diophantine equations we present several results on spectral characterizations of smith graphs

laplacian matrices and alexandrov topologies of digraphs abstract we explore the spectral properties of digraph laplacians and how they relate to topological properties of digraphs such as openness closure and strong connectedness under the alexandrov topology

spectral characterization of graphs with index at most 2 5 abstract a graph is said to be determined by the adjacency spectrum ds for short if there is no other nonisomorphic graph with the same spectrum all connected graphs with index at most 2 5 are known in this paper we show that with few exceptions all of these graphs are ds

a note on the spectral characterization of dumbbell graphs the dumbbell graph denoted by da b c is a bicyclic graph consisting of two vertex disjoint cycles ca and cb joined by a path pc 3 c 1 having only its end vertices in common with the two cycles by using a new cospectral invariant for r r 1 almost regular graphs we will show that almost all dumbbell graphs without cycle c4 as a subgraph are determined by the adjacency spectrum

laplacian coefficients of trees with given number of leaves or vertices of degree two abstract let g be a simple undirected n vertex graph with the characteristic polynomial of its laplacian matrix l g det i l g k 0 n 1 k c k n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of graph using a result of zhou and gutman on the relation between the laplacian coefficients and the matching numbers in subdivided bipartite graphs we characterize the trees with k leaves pendent vertices which simultaneously minimize all laplacian coefficients in particular this extremal balanced starlike tree s n k minimizes the wiener index the modified hyper wiener index and recently introduced laplacian like energy we prove that graph s n n 1 p has minimal laplacian coefficients among n vertex trees with p vertices of degree two in conclusion we illustrate on examples of these spectrum based invariants that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution and pose a conjecture on extremal unicyclic graphs with k leaves

on a laplacian spectral characterization of graphs of index less than 2 a graph is said to be determined by the adjacency respectively laplacian spectrum if there is no other non isomorphic graph with the same adjacency respectively laplacian spectrum the maximum eigenvalue of a g is called the index of g the connected graphs with index less than 2 are known and each is determined by its adjacency spectrum in this paper we show that graphs of index less than 2 are determined by their laplacian spectrum

on graphs with small number of laplacian eigenvalues greater than two all connected graphs with exactly one or two laplacian eigenvalues greater than two are determined

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

the characteristic polynomial of the laplacian of graphs in a b linear classes in this work we deal with the characteristic polynomial of the laplacian of a graph we present some general results about the coefficients of this polynomial we present families of graphs for which the number of edges m is given by a linear function of the number of vertices n in some of these graphs we can find certain coefficients of the above named polynomial as functions just of n

spectral characterization of graphs whose second largest eigenvalue is less than 1 abstract graphs with second largest eigenvalue 2 1 are extensively studied however whether they are determined by their adjacency spectra or not is less considered in this paper we completely characterize all the connected bipartite graphs with 2 1 that are determined by their adjacency spectra in addition we prove that all the connected non bipartite graphs with girth no less than 4 and 2 1 are determined by their adjacency spectra

spectral characterizations of lollipop graphs abstract the lollipop graph denoted by h n p is obtained by appending a cycle c p to a pendant vertex of a path p n p we will show that no two non isomorphic lollipop graphs are cospectral with respect to the adjacency matrix it is proved that for p odd the lollipop graphs h n p and some related graphs h n p are determined by the adjacency spectrum and that all lollipop graphs are determined by its laplacian spectrum

spectral characterization of unicyclic graphs whose second largest eigenvalue does not exceed 1 abstract connected graphs with the same number of vertices and edges are called unicyclic graphs all unicyclic graphs whose second largest eigenvalue does not exceed 1 denoted by u 2 are determined by xu 10 let g t s denote the graph obtained by attaching t 1 pendent paths of length 2 and s 0 pendent edges to a vertex of a triangle in this paper we first prove that the graph g t s is determined by its adjacency spectrum then we prove that the graphs in u 2 are determined by their adjacency spectra except for four graphs

multiplicity of integer roots of polynomials of graphs abstract let g be a graph with minimum vertex degree p 1 let b d a where d is the diagonal matrix of vertex degrees and a is the adjacency matrix of g the multiplicity of the integer root p of per xi b is characterized for bipartite graphs this characterization extends to per xi l where l d a is the laplacian matrix of g for graphs with unrestricted vertex degrees bounds are obtained on the multiplicities of integer roots of the permanental and characteristic polynomials of both l and b

recursive fermion system in cuntz algebra i embeddings of the car canonical anticommutation relations algebra of fermions into the cuntz algebra 2 or 2 d more generally are presented by using recursive constructions as a typical example an embedding of car onto the u 1 invariant subalgebra of 2 is constructed explicitly generalizing this construction to the case of 2 p an embedding of car onto the u 1 invariant subalgebra of 2 p is obtained restricting a permutation representation of the cuntz algebra we obtain the fock representation of car we apply the results to embed the algebra of parafermions of order p into 2 p according to green s ansatz

the cospectral equivalence classes of graphs having an isolated vertex let f g l be the characteristic polynomial of a graph g two graphs g and h are cospectral denoted by g h if f g l f h l by g f we denote the cospectral equivalence class determined by g under a graph g is said to be determined by its spectrum or simply g is a ds graph if h g whenever h g in this paper we determine the cospectral equivalence classes of three kinds of graphs having an isolated vertex find several ds graphs and identify the graph that has the fourth minimum index among all connected graphs with n vertices

on a signless laplacian spectral characterization of t shape trees let m be an associated matrix of a graph g the adjacency laplacian and signless laplacian matrix two graphs are said to be cospectral with respect to m if they have the same m spectrum a graph is said to be determined by m spectrum if there is no other non isomorphic graph with the same spectrum with respect to m it is shown that t shape trees are determined by their laplacian spectra moreover among them those are determined by their adjacency spectra are characterized in this paper we identify graphs which are cospectral to a given t shape tree with respect to the signless laplacian matrix subsequently t shape trees which are determined by their signless laplacian spectra are identified

which wheel graphs are determined by their laplacian spectra the wheel graph denoted by w n 1 is the graph obtained from the circuit c n with n vertices by adding a new vertex and joining it to every vertex of c n in this paper the wheel graph w n 1 except for w 7 is proved to be determined by its laplacian spectrum and a graph cospectral with the wheel graph w 7 is given

the spectral characterization of graphs of index less than 2 with no path as a component a graph is said to be determined by the adjacency and laplacian spectrum or to be a ds graph for short if there is no other non isomorphic graph with the same adjacency and laplacian spectrum respectively it is known that connected graphs of index less than 2 are determined by their adjacency spectrum in this paper we focus on the problem of characterization of ds graphs of index less than 2 first we give various infinite families of cospectral graphs with respect to the adjacency matrix subsequently the results will be used to characterize all ds graphs with respect to the adjacency matrix of index less than 2 with no path as a component moreover we show that most of these graphs are ds with respect to the laplacian matrix