stochastic and mixed flower graphs stochasticity is introduced to a well studied class of recursively grown graphs u v flower nets which have power law degree distributions as well as small world properties when u 1 the stochastic variant interpolates between different deterministic flower graphs thus adding flexibility to the model the random multiplicative growth process involved however leads to a spread ensemble of networks with finite variance for the number of links nodes and loops nevertheless the degree exponent and loopiness exponent attain unique values in the thermodynamic limit of infinitely large graphs we also study a class of mixed flower networks closely related to the stochastic flowers but which are grown recursively in a deterministic way the deterministic growth of mixed flower nets eliminates ensemble spreads and their recursive growth allows for exact analysis of their uniquely defined mixed properties

the mobius function of generalized subword order let p be a poset and let p be the set of all finite length words over p generalized subword order is the partial order on p obtained by letting u w if and only if there is a subword u of w having the same length as u such that each element of u is less than or equal to the corresponding element of u in the partial order on p classical subword order arises when p is an antichain while letting p be a chain gives an order on compositions for any finite poset p we give a simple formula for the mobius function of p in terms of the mobius function of p this permits us to rederive in an easy and uniform manner previous results of bjorner sagan and vatter and tomie we are also able to determine the homotopy type of all intervals in p for any finite p of rank at most 1

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

effect on normalized graph laplacian spectrum by motif attachment and duplication to some extent graph evolutionary mechanisms can be explained by its spectra here we are interested in two graph operations namely motif subgraph doubling and attachment that are biologically relevant we investigate how these two processes affect the spectrum of the normalized graph laplacian a high algebraic multiplicity of the eigenvalues 1 1 0 5 1 0 5 and others have been observed in the spectrum of many real networks we attempt to explain the production of distinct eigenvalues by motif doubling and attachment results on the eigenvalue 1 are discussed separately

anyonic chains topological defects and conformal field theory motivated by the three dimensional topological field theory two dimensional conformal field theory cft correspondence we study a broad class of one dimensional quantum mechanical models known as anyonic chains which can give rise to an enormously rich and largely unexplored space of two dimensional critical theories in the thermodynamic limit one remarkable feature of these systems is the appearance of non local microscopic topological symmetries that descend to topological defects of the resulting cfts we derive various model independent properties of these theories and of this topological symmetry topological defect correspondence for example by studying precursors of certain twist and defect fields directly in the anyonic chains we argue that under mild assumptions the two dimensional cfts correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras using this map we apply properties of defect hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories throughout we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones

eigenvalues for the transition matrix of a small world scale free network explicit expressions and applications the eigenvalues of the transition matrix for random walks on a network play a significant role in the structural and dynamical aspects of the network nevertheless it is still not well understood how the eigenvalues behave in small world and scale free networks which describe a large variety of real systems in this paper we study the eigenvalues for the transition matrix of a network that is simultaneously scale free small world and clustered we derive explicit simple expressions for all eigenvalues and their multiplicities with the spectral density exhibiting a power law form we then apply the obtained eigenvalues to determine the mixing time and random target access time for random walks both of which exhibit unusual behaviors compared with those for other networks signaling discernible effects of topologies on spectral features finally we use the eigenvalues to count spanning trees in the network

on the second largest normalized laplacian eigenvalue of graphs abstract let g v e be a simple graph of order n with normalized laplacian eigenvalues 1 2 n 1 n 0 the normalized laplacian spread of graph g denoted by 1 n 1 is the difference between the largest and the second smallest normalized laplacian eigenvalues of graph g in this paper we obtain the first four smallest values on 2 of graphs moreover we give a lower bound on 2 of connected bipartite graph g except the complete bipartite graph and characterize graphs for which the bound is attained finally we present some bounds on the normalized laplacian spread of graphs and characterize the extremal graphs

recursive graphs with small world scale free properties we discuss a category of graphs recursive clique trees which have small world and scale free properties and allow a fine tuning of the clustering and the power law exponent of their discrete degree distribution we determine relevant characteristics of those graphs the diameter degree distribution and clustering parameter the graphs have also an interesting recursive property and generalize recent constructions with fixed degree distributions

an iteration method for computing the total number of spanning trees and its applications in graph theory abstract calculating and analyzing the number of spanning trees of graphs network models is an important and interesting research project in wide variety of fields such as mathematics physics theoretical computer science chemistry and so on the number of spanning trees of graphs models displays amounts of information on its structural features and also on some relevant dynamical properties in particular network security random walks and percolation in this paper firstly due to lots of graphs models are built on the basis of various simple and small elements components we provide primarily some helpful network operation such as link operation and merge operation to generate more realistic and complicated graphs models secondly considering reliability of fault tolerance to random faults and of intrusion tolerance to selectively remove attacks synchronization capability and diffusion properties of networks we present an iterative method algorithm for computing the total number of spanning trees as a pellucid example we apply our method to tree space and cycle space notice that it is proved to be indeed a better tool in order to reflect more widely practical meanings we study its applications in graph theory including ladder graph with zero clustering coefficient wheel graph having nonzero clustering coefficient as constituent ingredients of maximal planar graphs in the rest of this paper we make a brief summary that the method described by us can be designed a program algorithm for obtaining easily the exact number of spanning trees of some models

random walks on weighted networks random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks in this article we study random walks on weighted networks with an arbitrary degree distribution where the weight of an edge between two nodes has a tunable parameter by using the spectral graph theory we derive analytical expressions for the stationary distribution mean first passage time mfpt average trapping time att and lower bound of the att which is defined as the average mfpt to a given node over every starting point chosen from the stationary distribution all these results depend on the weight parameter indicating a significant role of network weights on random walks for the case of uncorrelated networks we provide explicit formulas for the stationary distribution as well as att particularly for uncorrelated scale free networks when the target is placed on a node with the highest degree we show that att can display various scalings of network size depending also on the same parameter our findings could pave a way to delicately controlling random walk dynamics on complex networks

spectral analysis for weighted tree like fractals abstract much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized laplacian matrix in this paper we aim to present a study on the spectra of the normalized laplacian of weighted tree like fractals we analytically obtain the relationship between the eigenvalues and their multiplicities for two successive generations as an example of application of these results we then derive closed form expressions for their multiplicative kirchhoff index and kemeny s constant

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

discrete morse functions from lexicographic orders this paper shows how to construct a discrete morse function with a relatively small number of critical cells for the order complex of any finite poset with o and i from any lexicographic order on its maximal chains specifically if we attach facets according to the lexicographic order on maximal chains then each facet contributes at most one new face which is critical and at most one betti number changes facets which do not change the homotopy type also do not contribute any critical faces dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets as one application the mobius function may be computed as the alternating sum of morse numbers the above construction enables us to prove that the poset n s of partitions of a set 1 1 k k with repetition is homotopy equivalent to a wedge of spheres of top dimension when a is a hook shaped partition it is likely that the proof may be extended to a larger class of a and perhaps to all a despite a result of ziegler 1986 which shows that n s is not always cohen macaulay

calculating the normalized laplacian spectrum and the number of spanning trees of linear pentagonal chains abstract let w n be a linear pentagonal chain with 2 n pentagons in this article according to the decomposition theorem for the normalized laplacian polynomial of w n we obtain that the normalized laplacian spectrum of w n consists of the eigenvalues of two special matrices l a of order 3 n 1 and l s of order 2 n 1 together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices explicit closed form formulas for the degree kirchhoff index and the total number of spanning trees of w n are derived respectively finally it is interesting to find that the degree kirchhoff index of w n is approximately one half of its gutman index

exact evaluation of the causal spectrum and localization properties of electronic states on a scale free network abstract a deterministic network with tree structure is considered for which the spectrum of its adjacency matrix can be exactly evaluated by a recursive renormalization approach it amounts to successively increasing number of contributions at any finite step of construction of the tree resulting in a causal chain the resulting eigenvalues can be related the full energy spectrum of a nearest neighbor tight binding model defined on this structure given this association it turns out that further properties of the eigenvectors can be evaluated like the degree of quantum localization of the tight binding eigenstates expressed by the inverse participation ratio ipr it happens that for the current model the ipr s are also suitable to be analytically expressed in terms in corresponding eigenvalue chain the resulting ipr scaling behavior is expressed by the tails of eigenvalue chains as well

on the normalized laplacian energy and general randic index r 1 of graphs in this paper we consider the energy of a simple graph with respect to its normalized laplacian eigenvalues which we call the l energy over graphs of order n that contain no isolated vertices we characterize the graphs with minimal l energy of 2 and maximal l energy of 2 n 2 we provide upper and lower bounds for l energy based on its general randic index r 1 g we highlight known results for r 1 g most of which assume g is a tree we extend an upper bound of r 1 g known for trees to connected graphs we provide bounds on the l energy in terms of other parameters one of which is the energy with respect to the adjacency matrix finally we discuss the maximum change of l energy and r 1 g upon edge deletion

the normalized laplacians on both k triangle graph and k quadrilateral graph with their applications abstract the k triangle graph tk g is obtained from a graph g by replacing each edge in g with k 1 parallel paths in which one is of length 1 and each of the rest k paths is of length 2 whereas the k quadrilateral graph qk g is obtained from g by replacing each edge in g with k 1 parallel paths in which one is of length 1 and each of the rest k paths is of length 3 in this paper we completely determine the normalized laplacian spectrum on tk g resp qk g for any connected graph g k 2 as applications the correlation between the degree kirchhoff index the kemeny s constant and the number of spanning trees of tk g resp qk g the r th iterative k triangle graph t r k g the r th iterative k quadrilateral graph q r k g and those of g are derived our results extend those main results obtained in xie et al 2016 and li and hou 2017

the normalized laplacian spectrum of quadrilateral graphs and its applications abstract the quadrilateral graph q g of g is obtained from g by replacing each edge in g with two parallel paths of lengths 1 and 3 in this paper we completely describe the normalized laplacian spectrum on q g for any graph g as applications the significant formulae to calculate the multiplicative degree kirchhoff index the kemeny s constant and the number of spanning trees of q g and the quadrilateral iterative graph q r g are derived

the smallest eigenvalue for reversible markov chains we define a geometric quantity for reversible markov chains and use it to prove lower and upper bounds for the smallest eigenvalue of such chains known bounds are discussed and compared with the new ones

non local conserved charges from defects in perturbed conformal field theory perturbing a virasoro minimal model by the 1 3 primary bulk field results in an integrable field theory in this paper an infinite set of commuting conserved charges is obtained by considering defects a one parameter family of perturbed defect operators is given and it is shown that these operators mutually commute they commute with the hamiltonian of the perturbed conformal field theories and they satisfy a t system functional relation the formulation in terms of perturbed defects is a modification of the original prescription for such charges by bazhanov lukyanov and zamolodchikov

a general geometric growth model for pseudofractal scale free web abstract we propose a general geometric growth model for pseudofractal scale free web psw which is controlled by two tunable parameters we derive exactly the main characteristics of the networks degree distribution second moment of degree distribution degree correlations distribution of clustering coefficient as well as the diameter which are partially determined by the parameters analytical results show that the resulting networks are disassortative and follow power law degree distributions with a more general degree exponent tuned from 2 to 1 ln 3 ln 2 the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order the diameter grows logarithmically with the number of network nodes all these reveal that the networks described by our model have small world effect and scale free topology

eigentime identity of the weighted scale free triangulation networks for weight dependent walk abstract the eigenvalues of the normalized laplacian matrix of a network provide information on its structural properties and some relevant dynamical aspects in particular for weight dependent walk in order to get the eigentime identity for weight dependent walk we need to obtain the eigenvalues and their multiplicities of the laplacian matrix firstly the model of the weighted scale free triangulation networks is constructed then the eigenvalues and their multiplicities of transition weight matrix are presented after the recursive relationship of those eigenvalues at two successive generations are given consequently the laplacian spectrum is obtained finally the analytical expression of the eigentime identity indicating that the eigentime identity grows sublinearly with the network order is deduced

on the spectrum of the normalized laplacian of iterated triangulations of graphs the eigenvalues of the normalized laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects specifically in relation to random walks in this paper we determine the spectra of the normalized laplacian of iterated triangulations of a generic simple connected graph as an application we also find closed forms for their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

the normalized laplacian degree kirchhoff index and spanning trees of the linear polyomino chains let bn be a linear polyomino chain with n squares in this paper according to the decomposition theorem of normalized laplacian polynomial we obtain that the normalized laplacian spectrum of bn consists of the eigenvalues of two symmetric tridiagonal matrices of order n 1 together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices explicit closed formulas of the degree kirchhoff index and the number of spanning trees of bn are derived furthermore it is interesting to find that the degree kirchhoff index of bn is approximately one half of its gutman index

expected hitting times for random walks on the k triangle graph and their applications abstract given a simple connected graph g the k triangle graph of g written by tk g is obtained from g by adding k new vertices u i 1 u i 2 u i k for each edge e i u v in g and then adding in edges u u i 1 u u i 2 u u i k and u i 1 v u i 2 v u i k v in this paper the eigenvalues and eigenvectors of the probability transition matrix of random walks on tk g are completely determined then the expected hitting times between any two vertices of tk g are given in terms of those of g using these results all the relationship on the number of spanning trees resp kemeny s constant the degree kirchhoff index in tk g compared to those of g is found as well the resistance distance between any two vertices of tk g is given with respect to those of g

the mo bius function of factor order abstract intervals in the factor ordering of a free monoid are investigated it was shown by farmer 1982 that such intervals are contractible or homotopy spheres in case is the empty word we observe here that the same is true in general this implies that the mobius function of factor order takes values in 0 1 1 a recursive rule for this mobius function is given which allows efficient computation via the knuth morris pratt algorithm the mobius function of subword order was studied in bjorner 1990 we give here a simpler proof a parity changing involution for its combinatorial interpretation

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal