graphs and gromov hyperbolicity of non constant negatively curved surfaces in this paper we obtain the equivalence of the gromov hyperbolicity between an extensive class of complete riemannian surfaces with pinched negative curvature and certain kind of simple graphs whose edges have length 1 constructed following an easy triangular design of geodesics in the surface

a discrete nodal domain theorem for trees let g be a connected graph with n vertices and let x x1 xn be a real vector a positive negative sign graph of the vector x is a maximal connected subgraph of g on vertices xi 0 xi 0 for an eigenvalue of a generalized laplacian of a tree we characterize the maximal number of sign graphs of an eigenvector we give an o n2 time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an np complete problem author s abstract

metric spaces with linear extensions preserving lipschitz condition we study a new bi lipschitz invariant m of a metric space m its finiteness means that lipschitz functions on an arbitrary subset of m can be linearly extended to functions on m whose lipschitz constants are expanded by a factor controlled by m we prove that m is finite for several important classes of metric spaces these include metric trees of arbitrary cardinality groups of polynomial growth gromov hyperbolic groups certain classes of riemannian manifolds of bounded geometry and the finite direct sums of arbitrary combinations of these objects on the other hand we construct an example of a two dimensional riemannian manifold m of bounded geometry for which m 8

trees with a large laplacian eigenvalue multiplicity abstract in this paper we study the multiplicity of the laplacian eigenvalues of trees it is known that for trees integer laplacian eigenvalues larger than 1 are simple and also the multiplicity of laplacian eigenvalue 1 has been well studied before here we consider the multiplicities of the other non integral laplacian eigenvalues we give an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound n 3 2 and emphasize the particular role of the algebraic connectivity

some remarks on volume and diameter of riemannian manifolds soit m une variete compacte arbitraire et m i une famille de recouvrements finis de m s il existe c 0 tel que 1 m i c alors il existe des constantes positives a b et c telles que a log vol m i c diam m i b

anomalous diffusion of random walk on random planar maps we prove that the simple random walk on the uniform infinite planar triangulation uipt typically travels graph distance at most n 1 4 o n 1 in n units of time together with the complementary lower bound proven by gwynne and miller 2017 this shows that the typical graph distance displacement of the walk after n steps is n 1 4 o n 1 as conjectured by benjamini and curien geom funct anal 2 2 501 531 2013 arxiv 1202 5454 more generally we show that the simple random walks on a certain family of random planar maps in the gamma liouville quantum gravity lqg universality class for gamma in 0 2 including spanning tree weighted maps bipolar oriented maps and mated crt maps typically travels graph distance n 1 d gamma o n 1 in n units of time where d gamma is the growth exponent for the volume of a metric ball on the map which was shown to exist and depend only on gamma by ding and gwynne commun math phys 374 1877 1934 2018 arxiv 1807 01072 since d gamma 2 this shows that the simple random walk on each of these maps is subdiffusive our proofs are based on an embedding of the random planar maps under consideration into mathbb c wherein graph distance balls can be compared to euclidean balls modulo subpolynomial errors this embedding arises from a coupling of the given random planar map with a mated crt map together with the relationship of the latter map to sle decorated lqg

extending lipschitz functions via random metric partitions many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture inevitably the difficulty of such a procedure lies at the local boundary where overlapping views of the same locality must somehow be merged it is therefore desirable that the boundaries be smooth allowing a graceful transition from one viewpoint to the next for instance one may point to whitney s use of partitions of unity in studying what is now known as the whitney extension problem 36 37 in the present work we consider what is perhaps the most basic whitney type extension problem that of extending a lipschitz function so that it remains lipschitz often such a map is extended by first producing a cover of the new domain extending the mapping locally and then gluing together the individual pieces our main observation is that in many cases if one chooses a random cover from the right distribution the boundary can be made smooth on average even when the local maps are individually quite coarse this insight leads to the unification generalization and improvement of many known results as well as to new results for many interesting spaces

tree simplification and the plateaux phenomenon of graph laplacian eigenvalues we developed a procedure of reducing the number of vertices and edges of a given tree which we call the tree simplification procedure without changing its topological information our motivation for developing this procedure was to reduce computational costs of graph laplacian eigenvalues of such trees when we applied this procedure to a set of trees representing dendritic structures of retinal ganglion cells of a mouse and computed their graph laplacian eigenvalues we observed two plateaux i e two sets of multiple eigenvalues in the eigenvalue distribution of each such simplified tree in this article after describing our tree simplification procedure we analyze why such eigenvalue plateaux occur in a simplified tree explain such plateaux can occur in a more general graph if it satisfies a certain condition and identify these two eigenvalues specifically as well as the lower bound to their multiplicity

graph laplacians nodal domains and hyperplane arrangements eigenvectors of the laplacian of a graph g have received increasing attention in the recent past here we investigate their so called nodal domains i e the connected components of the maximal induced subgraphs of g on which an eigenvector psi does not change sign an analogue of courant s nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of psi in the spectrum this bound however is not sharp in general in this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph the class of boolean hypercubes is discussed in detail we find that despite the simplicity of this graph class for which complete spectral information is available the computations are still non trivial nevertheless we obtained some new results and a number of conjectures author s abstract

on the laplacian and signless laplacian spectrum of a graph with k pairwise co neighbor vertices abstract consider the laplacian and signless laplacian spectrum of a graph g of order n with k pairwise co neighbor vertices we prove that the number of shared neighbors is a laplacian and a signless laplacian eigenvalue of g with multiplicity at least k 1 additionally considering a connected graph g k with a vertex set defined by the k pairwise co neighbor vertices of g the laplacian spectrum of g k obtained from g adding the edges of g k includes l for each nonzero laplacian eigenvalue of g k the laplacian spectrum of g overlaps the laplacian spectrum of g k in at least n k 1 places

leaders in multi agent controllability under consensus algorithm and tree topology abstract by controllability of multi agent systems we mean the property that all agents can be driven to arbitrary configurations only through regulating a few leaders externally while the rest of the agents are subject to the nearest neighbor law it is known that the controllability of multi agent systems depends not only on the communication topologies among agents but also on the selection of leaders the main objective of this paper aims to characterize the virtue that leaders should have from the perspective of algebraic and graphical conditions we begin with a heuristic observation for tree graph to gain more insights necessary and sufficient conditions are then proposed in terms of downer branch and subgraphs respectively in particular it is shown for path topologies that controllability completely depends on the leaders location the obtained results may help us to pave the way towards a new method of formation and reconfiguration control in swarming robots

on the characterizing properties of the permanental polynomials of graphs abstract let g be a graph and a g the adjacency matrix of g the permanental polynomial of g is defined as g x per x i a g if two graphs g and h have the same permanental polynomial then g is called a per cospectral mate of h a graph g is said to be characterized by its permanental polynomial if all the per cospectral mates of g are isomorphic to g it is shown that complete graphs stars regular complete bipartite graphs and odd cycles are characterized by their permanental polynomials we prove that in general the permanental polynomial cannot characterize the paths and even cycles in particular for each l 1 and m 2 we can find non isomorphic per cospectral mates of p 4 l 3 and c 4 m respectively when we restrict our consideration to connected graphs both the paths and even cycles c 4 l 2 are characterized by their permanental polynomials

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

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 multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices abstract let g be a graph with adjacency matrix a g the nullity g of g is the multiplicity of zero as an eigenvalue of a g which has received a lot of attention because of its chemical importance the multiplicity of an arbitrary eigenvalue of a g is denoted as m g in 25 the authors proved that g 2 g p g for a connected graph g with equality if and only if g is a cycle of order divisible by 4 where g e g v g 1 is the cyclomatic number of g and p g is the number of pendant vertices of g in the present paper we intend to extend this result from the nullity of g to the multiplicity of an arbitrary eigenvalue of g differing from the method in 25 by mainly applying algebraic method we prove the following result for a connected graph g m g 2 g p g for an arbitrary eigenvalue of g the equality holds if and only if g is a cycle c n and 2 cos 2 k n with k 1 2 n 2 1

absolutely minimal lipschitz extension of tree valued mappings we prove that every lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal lipschitz extension amle we relate these extensions to a stochastic game called politics a generalization of a game called tug of war that has been used in peres et al j am math soc 22 1 167 210 2009 to study real valued amles

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

permanental polynomials of graphs abstract the first section surveys recent results on the permanental polynomial of a square matrix a i e per xi a the second section concerns the permanental polynomial of the adjacency matrix of a graph the final section is an introduction to the permanental polynomial of the laplacian matrix of a graph an appendix lists some of these latter polynomials

harmonicity and invariance on slices of the boolean cube in a recent work with kindler and wimmer we proved an invariance principle for the slice for low influence low degree harmonic multilinear polynomials a polynomial in x 1 ldots x n is harmonic if it is annihilated by sum i 1 n frac partial partial x i here we provide an alternative proof for general low degree harmonic multilinear polynomials with no constraints on the influences we show that any real valued harmonic multilinear polynomial on the slice whose degree is o sqrt n has approximately the same distribution under the slice and cube measures our proof is based on ideas and results from the representation theory of s n along with a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales while such decompositions have been used in the past for stationary reversible markov chains our decomposition is applied in a non stationary non reversible setup we also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof finally we provide independent simple proofs for the known facts that 1 one cannot distinguish between the slice and the cube based on functions of o n coordinates and 2 boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is o sqrt n

polyhedral object recognition by indexing in computer vision the indexing problem is the problem of recognizing a few objects in a large database of objects while avoiding the help of the classical image feature to object feature matching paradigm in this paper we address the problem of recognizing three dimensional 3 d polyhedral objects from 2 d images by indexing both the objects to be recognized and the images are represented by weighted graphs the indexing problem is therefore the problem of determining whether a graph extracted from the image is present or absent in a database of model graphs we introduce a novel method for performing this graph indexing process which is based both on polynomial characterization of binary and weighted graphs and on hashing we describe in detail this polynomial characterization and then we show how it can be used in the context of polyhedral object recognition next we describe a practical recognition by indexing system that includes the organization of the database the representation of polyhedral objects in terms of 2 d characteristic views the representation of this views in terms of weighted graphs and the associated image processing finally some experimental results allow the evaluation of the system performance

unveiling community structures in weighted networks random walks on simple graphs in connection with electrical resistor networks lead to the definition of markov chains with transition probability matrix in terms of electrical conductances we extend this definition to an effective transition matrix p ij to account for the probability of going from vertex i to any vertex j of the original connected graph g also we present an algorithm based on the definition of this effective transition matrix among vertices in the network to extract a topological feature related to the manner by which graph g has been organized this topological feature corresponds to the communities in the graph

vertex connectivity chromatic number domination number maximum degree and laplacian eigenvalue distribution abstract let g be a connected graph of order n and m g i be the number of laplacian eigenvalues of g in an interval i if i for a real number then m g is just the multiplicity of as a laplacian eigenvalue of g it is well known that the laplacian eigenvalues of g are all in the interval 0 n a lot of attention has been paid to the distribution of laplacian eigenvalues in the smallest subinterval 0 1 of length 1 in 0 n particularly hedetniemi et al 2016 14 proved that m g 0 1 if g has domination number we are interested in another extreme problem the distribution of laplacian eigenvalues in the largest subinterval n 1 n of length 1 in this article we prove that m g n 1 n and m g n 1 n 1 where and are respectively the vertex connectivity and the chromatic number of g two other main results of this paper focus on m g the multiplicity of an arbitrary laplacian eigenvalue of g it is proved that m g n and m g 1 n for a connected graph g with domination number and maximum degree

on metric ramsey type phenomena the main question studied in this article may be viewed as a nonlinear analogue of dvoretzky s theorem in banach space theory or as part of ramsey theory in combinatorics given a finite metric space on n points we seek its subspace of largest cardinality which can be embedded with a given distortion in hilbert space we provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion our main theorem states that for any 0 every n point metric space contains a subset of size at least n1 which is embeddable in hilbert space with o log 1 distortion the bound on the distortion is tight up to the log 1 factor we further include a comprehensive study of various other aspects of this problem

random walks on dynamical percolation mixing times mean squared displacement and hitting times we study the behavior of random walk on dynamical percolation in this model the edges of a graph g are either open or closed and refresh their status at rate mu while at the same time a random walker moves on g at rate 1 but only along edges which are open on the d dimensional torus with side length n we prove that in the subcritical regime the mixing times for both the full system and the random walker are n 2 mu up to constants we also obtain results concerning mean squared displacement and hitting times finally we show that the usual recurrence transience dichotomy for the lattice mathbb z d holds for this model as well

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

lipschitz quotients from metric trees and from banach spaces containing l1 abstract if x is any separable banach space containing l1 then there is a lipschitz quotient map from x onto any separable banach space y

on the geometry and laplacian of a graph abstract let g be a simple graph on n vertices and let l be the laplacian matrix of g we point out some connections between the geometric properties of g and the spectrum of l the multiplicities and eigenspaces as well as the eigenvalues of l are of geometric interest some historical information and relations of l to other matrices associated with g are also described

on one dimensional forward backward diffusion equations with linear convection and reaction abstract we study the initial neumann boundary value problem for a class of one dimensional forward backward diffusion equations with linear convection and reaction the diffusion flux function is assumed to contain two forward diffusion phases we prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases furthermore we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant the results are new even for the pure forward backward diffusion problem without convection and reaction our primary approach relies on a combination of the convex integration and baire s category methods for a related nonlocal differential inclusion

the uniform structure of banach spaces we explore the existence of uniformly continuous sections for quotient maps using this approach we are able to give a number of new examples in the theory of the uniform structure of banach spaces we show for example that there are two non isomorphic separable mathcal l 1 subspaces of l1 which are uniformly homeomorphic we also prove the existence of two coarsely homeomorphic banach spaces i e with lipschitz isomorphic nets which are not uniformly homeomorphic answering a question of johnson lindenstrauss and schechtman we construct a closed subspace of l1 whose unit ball is not an absolute uniform retract answering a question of the author

sharp uniform convexity and smoothness inequalities for trace norms we prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the schatten trace ideals c p which are the analogs of the lebesgue spaces l p in non commutative integration the inequalities are all precise analogs of results which had been known in l p but were only known in c p for special values of p in the course of our treatment of uniform convexity and smoothness inequalities for c p we obtain new and simple proofs of the known inequalities for l p

characterizations of almost surely continuous p stable random fourier series and strongly stationary processes on etend le probleme de dudley fernique aux processus p stables fortement stationnaires 1 p 2

proof of a conjecture on plateaux phenomenon of graph laplacian eigenvalues abstract let g be a simple graph a pendant path of g is a path such that one of its end vertices has degree 1 the other end has degree 3 and all the internal vertices have degree 2 let p k g be the number of pendant paths of length k of g and q k g be the number of vertices with degree 3 which are an end vertex of some pendant paths of length k motivated by the problem of characterizing dendritic trees n saito and e woei conjectured that any graph g has some laplacian eigenvalue with multiplicity at least p k g q k g we prove a more general result for both laplacian and signless laplacian eigenvalues from which the conjecture follows

on the multiplicity of laplacian eigenvalues and fiedler partitions abstract in this paper we investigate the relation between eigenvalue distribution and graph structure of two classes of graphs the m k stars and l dependent graphs we give conditions on the topology and edge weights in order to get values and multiplicities of laplacian matrix eigenvalues we prove that a vertex set reduction on graphs with m k star subgraphs is feasible keeping the same eigenvalues with reduced multiplicity moreover some useful eigenvectors properties are derived up to a product with a suitable matrix finally we relate these results with fiedler spectral partitioning of the graph and the physical relevance of the results is shortly discussed

on the bivariate permanent polynomials of graphs abstract in 1980 balasubramanian and parthasarathy introduced the bivariate permanent polynomials of graphs and conjectured that this polynomial is a graph characterizing polynomial that is any two graphs with the same bivariate permanent polynomial are isomorphic in this paper we give counterexamples to this conjecture by a computer search furthermore we show that several well known families of graphs are determined by the bivariate permanent polynomial complete graphs complete bipartite graphs regular complete multipartite graphs cycles and their complements

asymptotics of small eigenvalues of riemann surfaces on determine le comportement asymptotique precis des petites valeurs propres du laplacien sur une surface de riemann

markov chains in smooth banach spaces and gromov hyperbolic metric spaces a metric space x has markov type 2 if for any reversible flnite state markov chain fztg with z0 chosen according to the stationary distribution and any map f from the state space to x the distance dt from f z0 to f zt satisfles e d 2 k 2 te d 2 for some k k x 2 has markov type 2 this proves a conjecture of ball we also show that trees hyperbolic groups and simply connected riemannian manifolds of pinched negative curvature have markov type 2 our results are applied to settle several conjectures on lipschitz extensions and embeddings in particular we answer a question posed by johnson and lindenstrauss in 1982 by showing that for 1 q 2 p 1 any lipschitz mapping from a subset of lp to lq has a lipschitz extension deflned on all of lp

a bound for the permanent of the laplacian matrix abstract it is shown that if g is a simple connected graph on n vertices then per l g 2 n 1 g where l g is the laplacian matrix of g and g is the complexity of g

estimation of moments of sums of independent real random variables for the sum s sum x i of a sequence x i of independent symmetric or nonnegative random variables we give lower and upper estimates of moments of s the estimates are exact up to some universal constants and extend the previous results for particular types of variables x i

on the faria s inequality for the laplacian and signless laplacian spectra a unified approach abstract let p g and q g be the number of pendant vertices and quasi pendant vertices of a simple undirected graph g respectively let m l g 1 be the multiplicity of 1 as eigenvalue of a matrix which can be either the laplacian or the signless laplacian of a graph g a result due to i faria states that m l g 1 is bounded below by p g q g let r g be the number of internal vertices of g if r g q g following a unified approach we prove that m l g 1 p g q g if r g q g then we determine the equality m l g 1 p g q g m n 1 where m n 1 denotes the multiplicity of 1 as eigenvalue of a matrix n this matrix is obtained from either the laplacian or signless laplacian matrix of the subgraph induced by the internal vertices which are non quasi pendant vertices furthermore conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs

cat 0 spaces and expanders we construct a cat 0 space y with izeki nayatani invariant y 1 by a similar construction we also prove that there exists a cat 0 space which does not have markov type p for every p 1

edge perturbation on graphs with clusters adjacency laplacian and signless laplacian eigenvalues abstract let g be a simple undirected graph of order n a cluster in g of order c and degree s is a pair of vertex subsets c s where c is a set of cardinality c c 2 of pairwise co neighbor vertices sharing the same set s of s neighbors assuming that the graph g has k 1 clusters c 1 s 1 c k s k consider a family of k graphs h 1 h k and the graph g h 1 h k which is obtained from g after adding the edges of the graphs h 1 h k whose vertex set of each h j is identified with c j for j 1 k the laplacian eigenvalues of g h 1 h k remain the same independently of the graphs h 1 h k with the exception of c 1 c k k of them these new laplacian eigenvalues are determined using a unified approach which can also be applied to the determination of a same number of adjacency and signless laplacian eigenvalues when the graphs h 1 h k are regular the faria s lower bound on the multiplicity of the laplacian eigenvalue 1 of a graph with pendant vertices is generalized furthermore the algebraic connectivity and the laplacian index of g h 1 h k remain the same independently of the graphs h 1 h k

vector valued optimal lipschitz extensions consider a bounded open set and a lipschitz function does this function always have a canonical optimal lipschitz extension to all of u we propose a notion of optimal lipschitz extension and address existence and uniqueness in some special cases in the case n m 2 we show that smooth solutions have two phases in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans we also prove existence and uniqueness for the extension problem on finite graphs 2011 wiley periodicals inc

computing the permanental polynomials of graphs let m be an nn matrix with entries mij i j 1 2 n the permanent of m is defined to be per m i 1nmi i where the sum is taken over all permutations of 1 2 n the permanental polynomial of m is defined by per xinm where in is the identity matrix of size n in this paper we give recursive formulas for computing permanental polynomials of the laplacian matrix and the signless laplacian matrix of a graph respectively

a relation between the matching number and laplacian spectrum of a graph abstract let g be a graph its laplacian matrix is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix in this paper we generalize a result in r merris port math 48 3 1991 and obtain the following result let g be a graph and m g be a maximum matching in g then the number of edges in m g is a lower bound for the number of laplacian eigenvalues of g exceeding 2

unicyclic graphs with second largest and second smallest permanental sums abstract let a g be an adjacency matrix of a graph g then the polynomial g x per x i a g is called the permanental polynomial of g and the permanental sum of g is the sum of the absolute values of the coefficients of g x in this paper the second largest and second smallest permanental sums among connected unicyclic graphs and the corresponding extremal graphs are determined in addition we show that the computation of permanental sum is p complete

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

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

graph reduction techniques and the multiplicity of the laplacian eigenvalues abstract let m m i j be an n m real matrix be a nonzero real number and a be a symmetric real matrix we denote by d m the n n diagonal matrix d i a g j 1 m m 1 j j 1 m m n j and denote by l a the generalized laplacian matrix d a a a well known result of grone et al states that by connecting one of the end vertices of p 3 to an arbitrary vertex of a graph does not change the multiplicity of laplacian eigenvalue 1 we extend this theorem and some other results for a given generalized laplacian eigenvalue furthermore we give two proofs for a conjecture by saito and woei on the relation between the multiplicity of some laplacian eigenvalues and pendant paths

on the multiplicity of α as an eigenvalue of aα g of graphs with pendant vertices abstract let g be a simple undirected graph let 0 1 let a g d g 1 a g where d g and a g are the diagonal matrix of the vertex degrees of g and the adjacency matrix of g respectively let p g 0 and q g be the number of pendant vertices and quasi pendant vertices of g respectively let m g be the multiplicity of as an eigenvalue of a g it is proved that m g p g q g with equality if each internal vertex is a quasi pendant vertex if there is at least one internal vertex which is not a quasi pendant vertex the equality m g p g q g m n is determined in which m n is the multiplicity of as an eigenvalue of the matrix n this matrix is obtained from a g taking the entries corresponding to the internal vertices which are non quasi pendant vertices these results are applied to search for the multiplicity of as an eigenvalue of a g when g is a path a caterpillar a circular caterpillar a generalized bethe tree or a bethe tree for the bethe tree case a simple formula for the nullity is given

diffusive kpp equations with free boundaries in time almost periodic environments ii spreading speeds and semi wave solutions abstract in this series of papers we investigate the spreading and vanishing dynamics of time almost periodic diffusive kpp equations with free boundaries such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts in the first part of the series we showed that a spreading vanishing dichotomy occurs for such free boundary problems see 16 in this second part of the series we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs we first prove the existence of a unique time almost periodic semi wave solution associated to such a free boundary problem using the semi wave solution we then prove that the free boundary problem has a unique spreading speed

highly unique network descriptors based on the roots of the permanental polynomial in this paper we examine the zeros of permanental polynomials as highly unique network descriptors we employ exhaustively generated networks and demonstrate that our defined graph measures based on the moduli of the zeros of permanental polynomials are quite efficient when distinguishing graphs structurally in this work we continue with a line of research that relates to the search of almost complete graph invariants these highly unique network measures may serve as a powerful tool for tackling graph isomorphism

lp compression traveling salesmen and stable walks we show that if h is a group of polynomial growth whose growth rate is at least quadratic then the lp compression of the wreath product z oh equals max 1 p 1 2 we also show that the lp compression of z oz equals max p 2p 1 2 3 and the lp compression of z oz 0 the zero section of z oz equipped with the metric induced from z o z equals max p 1 2p 3 4 the fact that the hilbert compression exponent of z o z equals 3 while the hilbert compression exponent of z o z 0 equals 3 4 is used to show that there exists a lipschitz function f z o z 0 l2 which cannot be extended to a lipschitz function defined on all of z o z

continuum and lattice models analysis of the aggregation diffusion cell movement abstract the process by which one may take lattice models of a biophysical process and construct continuous models based on it is of mathematical interest as well as being of practical use such a process makes it possible to investigate how processes at the single cell level affect collective dynamics in this paper we first study the singular limit of a class of reinforced random walks on a lattice which simulates cell migration on discrete points and a complete analysis of the existence and stability of solutions is possible combined with the diffusion approximation assumption a continuous aggregation diffusion or backward forward parabolic equation is derived then the regularity estimate the nonexistence of the solution with small initial data and asymptotic behavior of the solution with large initial data are investigated in contrast to the continuous model the boundedness asymptotic behaviors of the lattice model solution in the diffusion region with monotone initial date and the interface behaviors of the aggregation diffusion regions are explored the numerical simulations agree well with theoretical results and biological observations