some bounds for total communicability of graphs abstract in a network or a graph the total communicability tc has been defined as the sum of the entries in the exponential of the adjacency matrix of the network this quantity offers a good measure of how easily information spreads across the network and can be useful in the design of networks having certain desirable properties in this paper we obtain some bounds for total communicability of a graph g t c g in terms of spectral radius of the adjacency matrix number of vertices number of edges minimum degree and the maximum degree of g moreover we find some upper bounds for t c g when g is the cartesian product tensor product or the strong product of two graphs in addition nordhaus gaddum type results for the total communicability of a graph g are established

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

perturbation results for the estrada index in weighted networks the logarithm of the estrada index has been proposed recently as a spectral measure to character efficiently the robustness of complex networks in this paper we explore the estrada index in weighted networks and develop various perturbation results based on spectral graph theory it is shown that the robustness of a network may be enhanced even when some edge weights are reduced this is of particular theoretical and practical significance to network design and optimization

performance analysis of the minimum variance beamformer we present an analysis of the signal to interference plus noise ratio sinr at the output of the minimum variance beamformer the analysis is based on the assumption that the signals and noise are gaussian and that the number of samples is large compared to the array size and it yields an explicit expression for the sinr in terms of the different parameters affecting the performance including signal to noise ratio snr interference to noise ratio inr signal to interference ratio sir angular separation between the desired signal and the interference array size and shape correlation between the desired signal and the interference and finite sample size

gaussianization of the spectra of graphs and networks theory and applications abstract matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks such as communicability node centrality bipartivity and many more they are also intimately related to the solution of differential equations describing dynamical processes on graphs and networks here we propose a new matrix function based on the gaussianization of the adjacency matrix of a graph this function gives more weight to a selected reference eigenvalue ref which may be located in any region of the graph spectra we show here that this matrix function can be derived from physical models that consider the interactions between nearest and next nearest neighbors in the graph we first obtain a few mathematical results for the trace of this matrix function when ref 1 h 1 for simple graphs as well as for random graphs we also provide a combinatorial interpretation of this index in terms of subgraphs in the graph and in terms of the competition pressure among agents in a complex system finally we apply this index to the study of magnetic properties of molecules emerging due to spin interactions as well as to studying the temporal evolution of the international trade network in the period 1992 2002 in both cases we give a clear phenomenological interpretation of the processes described

recursive updating the eigenvalue decomposition of a covariance matrix the author addresses the problem of computing the eigensystem of the modified hermitian matrix given the prior knowledge of the eigensystem of the original hermitian matrix specifically an additive rank k modification corresponding to adding and deleting blocks of data to and from the covariance matrix is considered an efficient and parallel algorithm which makes use of a generalized spectrum slicing theorem is derived for computing the eigenvalues the eigenvector can be computed explicitly in terms of the solution of a much reduced k k homogeneous hermitian system the overall computational complexity is shown to be improved by an order of magnitude from o n sup 3 to o n sup 2 k where n n is the size of the covariance matrix it is pointed out that these ideas can be applied to adaptive signal processing applications such as eigen based techniques for frequency or angle of arrival estimation and tracking specifically adaptive versions of the principal eigenvector method and the total least squares method are derived

on the extended estrada index of some graphs abstract let g be a graph on n vertices and 1 2 n the eigenvalues of its extended adjacency matrix the extended estrada index eeex is defined as the sum of the terms e i i 1 2 n in this paper we show the extended estrada index of the complete bipartite graphs and the friendship graphs also the relationship of the extended estrada index between the m splitting m shadow graphs and the original graphs are obtained

spectral moments of trees with given degree sequence abstract let 1 n be the eigenvalues of a graph g for any k 0 the k th spectral moment of g is defined by m k g 1 k n k we use the fact that m k g is also the number of closed walks of length k in g to show that among trees t whose degree sequence is d or majorized by d m k t is maximized by the greedy tree with degree sequence d constructed by assigning the highest degree in d to the root the second third highest degrees to the neighbors of the root and so on for any k 0 several corollaries follow in particular a conjecture of ilic and stevanovic on trees with given maximum degree which in turn implies a conjecture of gutman furtula markovic and glisic on the estrada index of such trees which is defined as ee g e 1 e n

on the estrada index conjecture abstract let g be a simple graph of order n with m edges let the adjacency spectrum be 1 2 n 1 n of g where 1 2 n 1 n the estrada index of a graph g is ee g i 1 n e i in j a pena i gutman j rada estimating the estrada index linear algebra appl 427 2007 70 76 pena et al posed a conjecture that the star s n has maximum estrada index for any tree of order n and the path p n has minimum estrada index for any tree of order n or any connected graph of order n in this paper we have proved that the star has maximum estrada index for any tree also we obtain that the path has minimum estrada index for any connected graph with m 1 8 n 4 or m n 2 6 moreover we give better lower bound on estrada index for any connected graph

information mobility in complex networks the concept of information mobility in complex networks is introduced on the basis of a stochastic process taking place in the network the transition matrix for this process represents the probability that the information arising at a given node is transferred to a target one we use the fractional powers of this transition matrix to investigate the stochastic process at fractional time intervals the mobility coefficient is then introduced on the basis of the trace of these fractional powers of the stochastic matrix the fractional time at which a network diffuses 50 of the information contained in its nodes 1 k50 is also introduced we then show that the scale free random networks display better spread of information than the non scale free ones we study 38 real world networks and analyze their performance in spreading information from their nodes we find that some real world networks perform even better than the scale free networks with the same average degree and we point out some of the structural parameters that make this possible

spectral measures of bipartivity in complex networks we introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity it is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks trophic interactions in food webs construction principles in metabolic networks or communities in social networks

on maximum estrada indices of k trees abstract the estrada index of a graph g is defined as ee g i 1 n e i where 1 2 n are the eigenvalues of the adjacency matrix of g let t n k be the set of the k trees of order n in this paper we characterize the graphs among t n k having the first resp the second largest estrada index

quadrature rule based bounds for functions of adjacency matrices bounds for entries of matrix functions based on gauss type quadrature rules are applied to adjacency matrices associated with graphs this technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities estrada index subgraph centrality communicability that describe properties of networks

accounting for the role of long walks on networks via a new matrix function abstract we introduce a new matrix function for studying graphs and real world networks based on a double factorial penalization of walks between nodes in a graph this new matrix function is based on the matrix error function we find a very good approximation of this function using a matrix hyperbolic tangent function we derive a communicability function a subgraph centrality and a double factorial estrada index based on this new matrix function we obtain upper and lower bounds for the double factorial estrada index of graphs showing that they are similar to those of the single factorial estrada index we then compare these indices with the single factorial one for simple graphs and real world networks we conclude that for networks containing chordless cycles holes the two penalization schemes produce significantly different results in particular we study two series of real world networks representing urban street networks and protein residue networks we observe that the subgraph centrality based on both indices produces a significantly different ranking of the nodes the use of the double factorial penalization of walks opens new possibilities for studying important structural properties of real world networks where long walks play a fundamental role such as the cases of networks containing chordless cycles

returnability in complex directed networks digraphs the concept of returnability is proposed for complex directed networks digraphs it can be seen as a generalization of the concept of reciprocity two measures of the returnability are introduced we establish closed formulas for the calculation of the returnability measures which are also related to the digraph spectrum the two measures are calculated for simple examples of digraphs as well as for real world complex directed networks and are compared with the reciprocity

a new proof of faber s intersection number conjecture we give a new proof of faber s intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves mg the proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles

the increase in the resolvent energy of a graph due to the addition of a new edge abstract the resolvent energy er g of a graph g on n vertices whose adjacency matrix has eigenvalues 1 n is the sum of the reciprocals of the numbers n 1 n n we introduce the resolvent energy matrix r g and present an algorithm that produces this matrix this algorithm may also be used to update r g when new edges are introduced to g using the resolvent energy matrix r g we determine the increase in the resolvent energy er g of g caused by such edge additions made to g moreover we express this increase in terms of the characteristic polynomial of g and the characteristic polynomials of three vertex deleted subgraphs of g

approximating spectral sums of large scale matrices using stochastic chebyshev approximations computation of the trace of a matrix function plays an important role in many scientific computing applications including applications in machine learning computational physics e g lattice qua

direction of arrival estimation by eigenstructure methods with imperfect spatial coherence of wave fronts eigenstructure subspace methods for direction of arrival estimation are only applicable when each wave front contributes a rank one component to the array covariance this implies that the wave fronts need to have perfect spatial coherence in several practical situations where the wave fronts may only have partial spatial coherence eigenstructure algorithms show poor performance such as overestimation of the number of sources bias and reduced accuracy and resolution in estimating the directions of arrival in this article the performance degradation due to partial spatial coherence is studied and a new approach is proposed that exploits the underlying coherence model to yield asymptotically exact estimates for finite data this method provides improved results only when the number of snapshots exceeds a certain threshold results of computer simulations are also presented

robust adaptive beamforming for general rank signal models the performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment sources or sensor array this is especially true when the desired signal components are present in the beamformer training data snapshots because in this case the adaptive array performance is very sensitive to array and model imperfections the similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly but the training sample size is small we propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary type mismatches of the desired signal array response our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank one point source and higher rank scattered source fluctuating wavefront desired signal models the proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst case performance optimization simple closed form solutions to the considered robust adaptive beamforming problems are derived our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms while at the same time offer a significantly improved robustness and faster convergence rates

maximum walk entropy implies walk regularity the notion of walk entropy sv g sv g for a graph g at the inverse temperature was put forward recently by estrada et al 2014 7 it was further proved by benzi 1 that a graph is walk regular if and only if its walk entropy is maximum for all temperatures i i where i is a set of real numbers containing at least an accumulation point benzi 1 conjectured that walk regularity can be characterized by the walk entropy if and only if there is a 0 0 such that sv g sv g is maximum here we prove that a graph is walk regular if and only if the view the mathml sourcesv g 1 lnn we also prove that if the graph is regular but not walk regular view the mathml sourcesv g 0 0 and view the mathml sourcelim 0sv g lnn lim sv g if the graph is not regular then view the mathml sourcesv g lnn for every 0 0 for some 0 0

a note on the graph s resolvent and the multifilar structure abstract we consider the recently discovered v ejov j a filar s k lucas p zograf clustering of spectra and fractals of regular graphs j math anal appl 333 2007 236 246 threadlike structure of the plot representing d regular graphs in the mean variance coordinates of exponential sums of the graph spectra in this note we demonstrate that this self similar phenomenon is more ubiquitous by exhibiting it with the help of a different generating function namely the mean and the variance of the resolvent of the adjacency matrix of the graph we also discuss the location of non hamiltonian graphs within this geometric structure

on maximum estrada indices of graphs with given parameters abstract for a graph g with eigenvalues 1 2 n its estrada index is defined as ee g i 1 n e i we determine the unique graphs with maximum estrada indices among graphs with given number of cut vertices connectivity and edge connectivity respectively

the estrada index of evolving graphs abstract the estrada index of a simple graph g with adjacency matrix a is defined as ee g trace e a it can be used in a range of fields as an efficient measuring tool an evolving graph g 1 g 2 g n is a time ordered sequence of graphs over a fixed set of vertices by using the natural definition of a walk on the evolving graph that respects the arrow of time we extend the static estrada index to accommodate this promising dynamic setting we find that although asymmetry is raised intrinsically by time s arrow the dynamic estrada index is order invariant we establish some lower and upper bounds for the dynamic estrada index in terms of the numbers of vertices and edges illustrative examples are worked out to demonstrate the computations involved

the estrada index of unicyclic graphs abstract the estrada index of a graph g is defined as ee g i 1 n e i where 1 2 n are the eigenvalues of g we determine the unique unicyclic graph with maximum estrada index the unique bipartite unicyclic graph with maximum estrada index and the unique bipartite unicyclic graph of a given bipartition with maximum estrada index we also show that only two graphs are possible to be the unicyclic graph s with minimum estrada index

estimating the estrada index abstract let g be a graph on n vertices and let 1 2 n be its eigenvalues the estrada index of g is a recently introduced graph invariant defined as ee i 1 n e i we establish lower and upper bounds for ee in terms of the number of vertices and number of edges also some inequalities between ee and the energy of g are obtained

on estrada index of trees abstract let g be a graph on n vertices and let 1 2 n be its eigenvalues the estrada index is defined as ee g i 1 n e i we determine the unique tree with maximum estrada index among the trees on n vertices with given matching number and the unique tree with maximum estrada index among the trees on n vertices with fixed diameter for n 1 3 we also determine the tree with maximum estrada index among the trees on n vertices with maximum degree it gives a partial solution to the conjecture proposed by ilic and stevanovic in ref 14

the estrada index of trees the estrada index of a graph g is defined as ee g i 1ne i where 1 2 n are the eigenvalues of its adjacency matrix we determine the unique tree with maximum estrada index among the set of trees with given number of pendant vertices as applications we determine trees with maximum estrada index among the set of trees with given matching number independence number and domination number respectively finally we give a proof of a conjecture in j li x li l wang the minimal estrada index of trees with two maximum degree vertices match commun math comput chem 64 2010 799 810 on trees with minimum estrada index among the set of trees with two adjacent vertices of maximum degree

bounds of the extended estrada index of graphs abstract let g be a graph on n vertices and 1 2 n the eigenvalues of its extended adjacency matrix the extended estrada index ee ex is defined as the sum of the terms e i i 1 2 n in this paper we establish lower and upper bounds for ee ex in terms of the number of vertices and the number of edges and characterize the extremal graphs also the bounds for ee ex of some special graphs are obtained

graphs with the maximal estrada indices abstract two new transformations are proposed to compare the estrada indices between two graphs let n m be the set of the n m graphs where n and m are the numbers of vertices and edges respectively the graphs with the maximal estrada indices in n m are deduced by the new method for three cases namely unicyclic and bipartite unicyclic graphs m n bicyclic graphs m n 1 and the n m graphs without even cycles n 1 m 3 n 1 2