Spectral Graph Theory

The $N$-particle Schr\"odinger operator $H(K),$ $K\in (-\pi,\pi]^d,$ $K$ being the total quasi-momentum, with short-range pair potentials on lattice $\mathbb{Z}^d,$ $d\ge1,$ is considered. For fixed total quasi-momentum $K$, the structure of $H(K)$'s essential spectrum is described and the analogue of the Hunziker -- van Winter -- Zhislin (HVZ) theorem is proved.

In this paper, we deal with the spectrum of line graph using Signless Laplacian matrix. We consider the cases of complete, regular bipartite and regular graph and justify our results by taking examples over it. We derive some generalized results about spectrum, rank, nullity of the line graph on the basis of Signless Laplacian matrix and observations on it.

In this paper, we utilize the signless Laplacian eigenvalues of line graphs and the dominant number to determine the limits of chromatic numbers for the line graphs of certain special graphs. The application of Vizing's theorem helps to justify the existence of larger boundaries in this particular case.

Let G be a simple connected graph with degree sequence (d 1 , d 2 , . . . , dn) i=1 µ i denote the Kirchhoff index and the number of spanning trees of G, respectively. In this paper we establish several lower bounds for Kf (G) in terms of τ (G), the order, the size and maximum degree of G.

Let G = (V, E), V = {1, 2, . . . , n}, be a simple connected graph of order n and size m, with sequence of vertex degrees the Kirchhoff index and the number of spanning trees of G, respectively. In this paper we determine several lower bounds for K f (G) depending on t(G) and some of the graph parameters n, m or ∆.

The resolvent energy of a graph G of order n is defined as the eigenvalues of G. Lower and upper bounds for the resolvent energy of a graph, which depend on some of the parameters n, λ 1 , λ n , det(R A (n)) = n ∏ i=1 1 n-λ i , are obtained.

Let G be a simple connected graph of order n, sequence of vertex degrees In this paper we determine several lower and upper bounds for K f depending on some of the graph parameters such as number of vertices, maximum degree, minimum degree, and number of spanning trees or multiplicative Zagreb index.

For a simple connected graph G of order n and size m, the Laplacian energy of G is defined as In this note, some new lower bounds on the graph invariant LE(G) are derived. The obtained results are compared with some already known lower bounds of LE(G).

Let G be an undirected connected graph with n, n ≥ 3, vertices and m edges. If 0 are the Laplacian and the normalized Laplacian eigenvalues of G, then the Kirchhoff and the degree Kirchhoff indices obey the relations K f i , respectively. The inequalities that determine lower bounds for some invariants of G, that contain K f (G) and DK f (G) , are obtained in this paper. Lower bounds for K f (G) and DK f (G), known in the literature, are obtained as a special case.

Let Gσ be a graph obtained by attaching a self-loop, or just a loop, for short, at each of σ chosen vertices of a given graph G. Gutman et al. have recently introduced the concept of the energy of graphs with self-loops, and conjectured that the energy E(G) of a graph G of order n is always strictly less than the energy E(Gσ) of a corresponding graph Gσ, for 1 ≤ σ ≤ n − 1. In this paper, a simple set of graphs which disproves this conjecture is exposed, together with some remarks regarding the standard deviations of the (adjacency) eigenvalues of G and Gσ, respectively.

Let G be a simple connected graph of order n, sequence of vertex degrees In this paper we determine several lower and upper bounds for K f depending on some of the graph parameters such as number of vertices, maximum degree, minimum degree, and number of spanning trees or multiplicative Zagreb index.

c © 2019 the authors. This is an open access article under the CC BY (International 4.0) license (https://creativecommons.org/licenses/by/4.0/). Abstract LetG be a simple graph of order n and letL be its Laplacian matrix. Eigenvalues of the matrixL are denoted by μ1, μ2, · · · , μn and it is assumed that μ1 > μ2 > · · · > μn. The Laplacian resolvent energy and Kirchhoff index of the graph G are defined as RL(G) = ∑n i=1 1 n+1−μi and Kf(G) = n ∑n−1 i=1 1 μi , respectively. In this paper, we derive some bounds on the invariant RL(G) and establish a relation between RL(G) and Kf(G).

Let G = (V, E), V = {1, 2, . . . , n}, be a simple graph without isolated vertices, with vertex degree sequence is measure of irregularity of graph G with the property I(G) = 0 if and only if G is regular, and I(G) > 0 otherwise. In this paper we introduce some new irregularity measures.

Let G be an undirected connected graph with n, n 3, vertices and m edges with Laplacian eigenvalues µ 1 µ 2 . . . µ n-1 > µn = 0. Denote by µ I = µr 1 + µr 2 + . . . + µr k , 1 k n -2, 1 r 1 < r 2 < . . . < r k n -1, the sum of k arbitrary Laplacian eigenvalues, with µ I1 = µ 1 + µ 2 + . . . + µ k and µ In = µ n-k + . . . + µ n-1 . Lower bounds of graph invariants µ I1 -µ In and µ I1 /µ In are obtained. Some known inequalities follow as a special case.

We investigate the structural patterns of the appearance and disappearance of links in dynamic knowledge networks. Human knowledge is nowadays increasingly created and curated online, in a collaborative and highly dynamic fashion. The knowledge thus created is interlinked in nature, and an important open task is to understand its temporal evolution. In this paper, we study the underlying mechanisms of changes in knowledge networks which are of structural nature, i.e., which are a direct result of a knowledge network's structure. Concretely, we ask whether the appearance and disappearance of interconnections between concepts (items of a knowledge base) can be predicted using information about the network formed by these interconnections. In contrast to related work on this problem, we take into account the disappearance of links in our study, to account for the fact that the evolution of collaborative knowledge bases includes a high proportion of removals and reverts. We perform an empirical study on the best-known and largest collaborative knowledge base, Wikipedia, and show that traditional indicators of structural change used in the link analysis literature can be classified into four classes, which we show to indicate growth, decay, stability and instability of links. We finally use these methods to identify the underlying reasons for individual additions and removals of knowledge links.

We investigate the structural patterns of the appearance and disappearance of links in dynamic knowledge networks. Human knowledge is nowadays increasingly created and curated online, in a collaborative and highly dynamic fashion. The knowledge thus created is interlinked in nature, and an important open task is to understand its temporal evolution. In this paper, we study the underlying mechanisms of changes in knowledge networks which are of structural nature, i.e., which are a direct result of a knowledge network's structure. Concretely, we ask whether the appearance and disappearance of interconnections between concepts (items of a knowledge base) can be predicted using information about the network formed by these interconnections. In contrast to related work on this problem, we take into account the disappearance of links in our study, to account for the fact that the evolution of collaborative knowledge bases includes a high proportion of removals and reverts. We perform an empirical study on the best-known and largest collaborative knowledge base, Wikipedia, and show that traditional indicators of structural change used in the link analysis literature can be classified into four classes, which we show to indicate growth, decay, stability and instability of links. We finally use these methods to identify the underlying reasons for individual additions and removals of knowledge links.

The co-prime order graph of a group is a simple finite graph having vertex set as group itself, and there is an edge between vertex u to the other vertex v whenever gcd(o(u), o(v)) = 1 or prime. Spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, maximum degree matrix, minimum degree matrix, etc. In this article, we have extended the study from the Laplacian matrix spectrum to various degree based matrices such as signless Laplacian matrix, normalized Laplacian matrix, maximum degree and minimum degree matrix of co-prime order graph of finite groups Z t p , Z p α 1 × Z p α 2 ×. .. × Z p αn , Z p t × Z q s ,D p n and also provided some general results for any finite group. Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix are based on adjacency matrix and degree matrix. The maximum degree matrix is based on the maximum of degree of a vertex among all the pairs of vertices whereas the minimum degree matrix is based on the minimum of degree of a vertex among all the pairs of vertices.

Call a function f : F n 2 → {0, 1} odd-cycle-free if there are no x 1 , . . . , x k ∈ F n 2 with k an odd integer such that f We show that one can distinguish odd-cycle-free functions from those ǫ-far from being odd-cycle-free by making poly(1/ǫ) queries to an evaluation oracle. To obtain this result, we use connections between basic Fourier analysis and spectral graph theory to show that one can reduce testing odd-cyclefreeness of Boolean functions to testing bipartiteness of dense graphs. Our work forms part of a recent sequence of works that shows connections between testability of properties of Boolean functions and of graph properties. We also prove that there is a canonical tester for odd-cycle-freeness making poly(1/ǫ) queries, meaning that the testing algorithm operates by picking a random linear subspace of dimension O(log 1/ǫ) and then checking if the restriction of the function to the subspace is odd-cycle-free or not. The test is analyzed by studying the effect of random subspace restriction on the Fourier coefficients of a function. Our work implies that testing odd-cycle-freeness using a canonical tester instead of an arbitrary tester incurs no more than a polynomial blowup in the query complexity. The question of whether a canonical tester with polynomial blowup exists for all linear-invariant properties remains an open problem.

In Graph Signal Processing (GSP), data dependencies are represented by a graph whose nodes label the data and the edges capture dependencies among nodes. The graph is represented by a weighted adjacency matrix A that, in GSP, generalizes the Discrete Signal Processing (DSP) shift operator z -1 . The (right) eigenvectors of the shift A (graph spectral components) diagonalize A and lead to a graph Fourier basis F that provides a graph spectral representation of the graph signal. The inverse of the (matrix of the) graph Fourier basis F is the Graph Fourier transform (GFT), F -1 . Often, including in real world examples, this diagonalization is numerically unstable. This paper develops an approach to compute an accurate approximation to F and F -1 , while insuring their numerical stability, by means of solving a non convex optimization problem. To address the nonconvexity, we propose an algorithm, the stable graph Fourier basis algorithm (SGFA) that we prove to exponentially increase the accuracy of the approximating F per iteration. Likewise, we can apply SGFA to A H and, hence, approximate the stable left eigenvectors for the graph shift A and directly compute the GFT. We evaluate empirically the quality of SGFA by applying it to graph shifts A drawn from two real world problems, the 2004 US political blogs graph and the Manhattan road map, carrying out a comprehensive study on tradeoffs between different SGFA parameters. We also confirm our conclusions by applying SGFA on very sparse and very dense directed Erdős-Rényi graphs.

We study the possibility of the existence of a Katona type proof for the Erdős-Ko-Rado theorem for 2-and 3-intersecting families of sets. An Erdős-Ko-Rado type theorem for 2-intersecting integer arithmetic progressions and a model theoretic argument show that such an approach works in the 2-intersecting case, at least for some values of n and k.

We construct a compact integral operator K z on L 2 (0, ∞), we prove det(1-K z) = ξ(s)/ξ(1-s), and then via cluster expansion, Borel convergence and OS-reflection-positivity we recover a self-adjoint "Hilbert-Pólya" operator, whose eigenvalues correspond to the zeros of the Riemann zeta function, which implies ℜs = 1 2 .

Let Gσ be the graph obtained from a simple graph G of order n by adding σ self-loops, one self-loop at each vertex in S ⊆ V (G). Let λ1(Gσ), λ2(Gσ), . . . , λn(Gσ) be the eigenvalues of Gσ. The energy of Gσ, denoted by E (Gσ), is defined as E (Gσ) = n i=1 λi(Gσ) -σ n . In this paper, using various analytic inequalities and previously established results, we derive several new lower and upper bounds on E (Gσ).

This article introduces a groundbreaking formalism for color, not as a perceptual byproduct of wavelength, but as a phase-admitted structure, governed by the scalar function φₑ(x, t, λ). We propose a harmonic framework where the average spectral wavelength λ₀ = 537.5 nm serves as a color tuning fork, and each perceived hue becomes a spectral tone manifesting only under conditions of phase permissibility. Further, we define the Ω₁ unit-an analogue to the musical 1 Hz-enabling micro-modulation of color across nonlinear gradients. The culmination of this framework is the Higgs Phase Swing Equation, uniting mass, color, and perceptual admissibility into a singular model. 0 1 F 5 3 8 Section 1: Spectral Admissibility Function Let φₑ(x, t, λ) be the scalar function defining whether a color of wavelength λ is permitted to manifest within a perceptual or environmental phase field at spacetime coordinates (x, t). Color is admitted, not inherent. It emerges only when: δₘ(x, t, λ) = +1 if φₑ(x, t, λ) ≥ ε; otherwise δ 2 0 9 8 = 0 where ε is the threshold of admissibility. 0 1 F 5 3 8 Section 2: Phase-Spectral Keyboard-λ₀ as Harmonic Anchor We establish λ₀ = 537.5 nm as the average of the seven principal colors, corresponding to green-yellow and becoming our zero-tone. Following musical structure: λ 2 0 9 9 = λ₀ • 2^(n/12) where n is the number of tonal steps (-6 to +6), forming a 13-tone spectral scale. The 13th tone, λ₁₃ = λ₀ • 2^(13/12), symbolizes a spectral phase beyond admissible sight-akin to the "shadow note".

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic graphs is analyzed and they are characterized in terms of their main eigenvalues without any restriction on its combinatorial structure. We give a necessary and sufficient condition for a graph G to have -1λ min as an eigenvalue of its complement, where λ min denotes the least eigenvalue of G. Also, we prove that among connected bipartite graphs, K r,r is the unique graph for which the index of the complement is equal to -1λ min . Finally, we characterize all paths and all double stars (trees with diameter three) for which the smallest eigenvalue is non-main. Main eigenvalues of paths and double stars are identified.

Let m G (I) denote the number of Laplacian eigenvalues of a graph G in an interval I, and let γ(G) denote its domination number. We extend the recent result m G [0, 1) ≤ γ(G), and show that isolate-free graphs also satisfy γ(G) ≤ m G [2, n]. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of γ(G), showing that γ(G) mG[0,1) ∈ O(log n).

A weighted Bethe graph B is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family F of graphs. The operation of identifying the root vertex of each of r weighted Bethe graphs to the vertices of a connected graph R of order r is introduced as the R-concatenation of a family of r weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when F has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in F are all regular) of the R-concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

In the set of all connected graphs with fixed order and size, the graphs with maximal index are nested split graphs, also called threshold graphs. It was recently (and independently) observed in [F.K.Bell, D. Cvetković, P. Rowlinson, S.K. Simić, Graphs for which the largest eigenvalue is minimal, II, Linear Algebra Appl. 429 (2008)] and [A. Bhattacharya, S. Friedland, U.N. Peled, On the first eigenvalue of bipartite graphs, Electron. J. Combin. 15 (2008), #144] that double nested graphs, also called bipartite chain graphs, play the same role within class of bipartite graphs. In this paper we study some structural and spectral features of double nested graphs. In studying the spectrum of double nested graphs we rather consider some weighted nonnegative matrices (of significantly less order) which preserve all positive eigenvalues of former ones. Moreover, their inverse matrices appear to be tridiagonal. Using this fact we provide several new bounds on the index (largest eigenvalue) of double nested graphs, and also deduce some bounds on eigenvector components for the index. We conclude the paper by examining the questions related to main versus non-main eigenvalues.

Considering a graph H of order p, a generalized H-join operation of a family of graphs G1, . . . , Gp, constrained by a family of vertex subsets Si ⊆ V (Gi), i = 1, . . . , p, is introduced. When each vertex subset Si is (ki, τi)-regular, it is deduced that all non-main adjacency eigenvalues of Gi, different from kiτi, for i = 1, . . . , p, remain as eigenvalues of the graph G obtained by the above mentioned operation. Furthermore, if each graph Gi of the family is ki-regular, for i = 1, . . . , p, and all the vertex subsets are such that Si = V (Gi), the H-generalized join operation constrained by these vertex subsets coincides with the H-generalized join operation. Some applications on the spread of graphs are presented. Namely, new lower and upper bounds are deduced and a infinity family of non regular graphs of order n with spread equals n is introduced.

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved in [MATCH Commun. Math. Comput. Chem. 79 (2018) for every bipartite graph G of order n, size m and maximum degree ∆. We prove the above bound for all graphs G. We also prove new types of two bounds of Koolen and Moulton given in [Adv. Appl. Math. 26 (2001) 47-52] and [Graphs Comb. 19 (2003) 131-135].

Feature selection aims to reduce dimensionality for building comprehensible learning models with good generalization performance. Feature selection algorithms are largely studied separately according to the type of learning: supervised or unsupervised. This work exploits intrinsic properties underlying supervised and unsupervised feature selection algorithms, and proposes a unified framework for feature selection based on spectral graph theory. The proposed framework is able to generate families of algorithms for both supervised and unsupervised feature selection. And we show that existing powerful algorithms such as ReliefF (supervised) and Laplacian Score (unsupervised) are special cases of the proposed framework. To the best of our knowledge, this work is the first attempt to unify supervised and unsupervised feature selection, and enable their joint study under a general framework. Experiments demonstrated the efficacy of the novel algorithms derived from the framework.

In this paper we propose a distributed algorithm for the estimation and control of the connectivity of ad-hoc networks in the presence of a random topology. First, given a generic random graph, we introduce a novel stochastic power iteration method that allows each node to estimate and track the algebraic connectivity of the underlying expected graph. Using results from stochastic approximation theory, we prove that the proposed method converges almost surely (a.s.) to the desired value of connectivity even in the presence of imperfect communication scenarios. The estimation strategy is then used as a basic tool to adapt the power transmitted by each node of a wireless network, in order to maximize the network connectivity in the presence of realistic Medium Access Control (MAC) protocols or simply to drive the connectivity toward a desired target value. Numerical results corroborate our theoretical findings, thus illustrating the main features of the algorithm and its robustness to fluctuations of the network graph due to the presence of random link failures.

In this paper we propose a distributed algorithm for the estimation and control of the connectivity of ad-hoc networks in the presence of a random topology. First, given a generic random graph, we introduce a novel stochastic power iteration method that allows each node to estimate and track the algebraic connectivity of the underlying expected graph. Using results from stochastic approximation theory, we prove that the proposed method converges almost surely (a.s.) to the desired value of connectivity even in the presence of imperfect communication scenarios. The estimation strategy is then used as a basic tool to adapt the power transmitted by each node of a wireless network, in order to maximize the network connectivity in the presence of realistic Medium Access Control (MAC) protocols or simply to drive the connectivity toward a desired target value. Numerical results corroborate our theoretical findings, thus illustrating the main features of the algorithm and its robustness to fluctuations of the network graph due to the presence of random link failures.

Resumo: O Projecto Okutanga surge como uma iniciativa de grande relevância, idealizada pelo Instituto Superior Politécnico Ndunduma (ISPN). Alinhado com uma visão humanista e focado na promoção da dignidade humana através da educação, o Projecto visa a alfabetização de mulheres zungueiras da cidade de Cuito, na província do Bié, Angola. Essas mulheres, que trabalham no comércio informal, frequentemente enfrentam barreiras socioeconômicas e culturais que limitam seu acesso à educação formal, resultando em uma exclusão significativa do sistema educacional e, consequentemente, de oportunidades de melhoria em suas condições de vida. Este estudo busca analisar o impacto desse Projecto, enfatizando sua importância para a inclusão social e econômica dessas mulheres. Através de uma abordagem qualitativa, com entrevistas semiestruturadas e observação participante, os dados obtidos revelam benefícios substanciais. As participantes relataram um aumento na autoestima, especialmente pela capacidade recém-adquirida de ler e escrever, além de melhorias em suas oportunidades econômicas, como o manejo mais eficiente de seus pequenos negócios. O Projecto se consolida como uma ferramenta essencial para o empoderamento feminino, proporcionando a essas mulheres maior autonomia e reconhecimento dentro de suas comunidades. Ademais, o Projecto Okutanga contribui de maneira significativa para a redução da desigualdade social na região, evidenciando o poder transformador da educação na vida dessas mulheres e em suas famílias.

A graph (finite and without loops or multiple edges) is geodetic if there is at most one shortest path between each pair of vertices. The author proves the Theorem: If G is a connected graph with at least one edge, then the line graph of G is geodetic if and only if G is a tree or an odd cycle. Similarly, the middle graph of a connected graph is geodetic if and only if G is a tree, while the total graph of a graph is geodetic if and only if every connected component of G has at most one edge. The author also defines weakly and strongly geodetic graphs and gives the theorems corresponding to those above.

In this note, we consider G to be an undirected bipartite graph with n vertices and e edges. We present some new upper and lower bounds for eigenvalues and energy of G in terms of e and Ni ∩ Nj (number of common vertices adjacent to vi and vj for i, j = 1,. .. , p) where, Ni denote the set of neighbours of vertices vi(i = 1,. .. , p).

This paper looks at the task of network topology inference, where the goal is to learn an unknown graph from nodal observations. One of the novelties of the approach put forth is the consideration of prior information about the density of motifs of the unknown graph to enhance the inference of classical Gaussian graphical models. Dealing with the density of motifs directly constitutes a challenging combinatorial task. However, we note that if two graphs have similar motif densities, one can show that the expected value of a polynomial applied to their empirical spectral distributions will be similar. Guided by this, we first assume that we have a reference graph that is related to the sought graph (in the sense of having similar motif densities) and then, we exploit this relation by incorporating a similarity constraint and a regularization term in the network topology inference optimization problem. The (non-)convexity of the optimization problem is discussed and a computational efficient alternating majorizationminimization algorithm is designed. We assess the performance of the proposed method through exhaustive numerical experiments where different constraints are considered and compared against popular baselines algorithms on both synthetic and real-world datasets.

We address the problem of identifying a graph structure from the observation of signals defined on its nodes. Fundamentally, the unknown graph encodes direct relationships between signal elements, which we aim to recover from observable indirect relationships generated by a diffusion process on the graph. The fresh look advocated here permeates benefits from convex optimization and stationarity of graph signals, in order to identify the graph shift operator (a matrix representation of the graph) given only its eigenvectors. These spectral templates can be obtained, e.g., from the sample covariance of independent graph signals diffused on the sought network. The novel idea is to find a graph shift that, while being consistent with the provided spectral information, endows the network with certain desired properties such as sparsity. To that end we develop efficient inference algorithms stemming from provably-tight convex relaxations of natural nonconvex criteria, particularizing the results for two shifts: the adjacency matrix and the normalized Laplacian. Algorithms and theoretical recovery conditions are developed not only when the templates are perfectly known, but also when the eigenvectors are noisy or when only a subset of them are given. Numerical tests showcase the effectiveness of the proposed algorithms in recovering social, brain, and amino-acid networks.

An analytical algebraic approach for distributed network identification is presented in this paper. The information propagation in the network is modeled using a state-space representation. Using the observations recorded at a single node and a known excitation signal, we present algorithms to compute the eigenfrequencies and eigenmodes of the graph in a distributed manner. The eigenfrequencies of the graph may be computed using a generalized eigenvalue algorithm, while the eigenmodes can be computed using an eigenvalue decomposition. The developed theory is demonstrated using numerical experiments.

Softwaredefined networks (SDN) are an emerging technology that offers to simplify networking devices by centralizing the network layer functions and allowing adaptively programmable traffic flows. We propose using spectral graph theory methods to identify and locate congestion in a network. The analysis of the balanced traffic case yields an efficient solution for congestion identification. The unbalanced case demonstrates a distinct drop in connectivity that can be used to determine the onset of congestion. The eigenvectors of the Laplacian matrix are used to locate the congestion and achieve effective graph partitioning.

The study of type III RNases constitutes an important area in molecular biology. It is known that the pac1 + gene encodes a particular RNase III that shares low amino acid similarity with other genes despite having a double-stranded ribonuclease activity. Bioinformatics methods based on sequence alignment may fail when there is a low amino acidic identity percentage between query sequence and others with similar functions (remote homologues) or a similar sequence is not recorded in the database. Quantitative Structure-Activity Relationships (QSAR) applied to protein sequences may allow an alignment-independent prediction of protein function. These sequences QSAR like methods often use 1D sequence numerical parameters as the input to seek sequence-function relationships. However, previous 2D representation of sequences may uncover useful higher-order information. In the work described here we calculated for the first time the Spectral Moments of a Markov Matrix (MMM) associated with a 2D-HP-map of a protein sequence. We used MMMs values to characterize numerically 81 sequences of type III RNases and 133 proteins of a control group. We subsequently developed one MMM-QSAR and one classic Hidden Markov Model (HMM) based on the same data. The MMM-QSAR showed a discrimination power of RNAses from other proteins of 97.35% without using alignment, which is a result as good as for the known HMM techniques. We also report for the first time the isolation of a new Pac1 protein (DQ647826) from Schizosaccharomyces pombe, strain 428-4-1. The MMM-QSAR model predicts the new RNase III with the same accuracy as other classical alignment methods. Experimental assay of this protein confirms the predicted activity. The present results suggest that MMM-QSAR models may be used for protein function annotation avoiding sequence alignment with the same accuracy of classic HMM models.

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and singless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy. AMS classification: 5C50, 15A48 Graph spectrum, Laplacian graph spectrum, signless Laplacina spectrum, Laplacian energy, signless Laplacian energy. 1 Preliminaries In this paper we consider simple unordered graphs G, herein just called graphs, with vertex set V(G) and edge set E(G). We s...

The Gallai fuzzy graph Γ(G) of a fuzzy graph G has the fuzzy edges of G as its fuzzy vertices and two distinct fuzzy edges of G are fuzzy incident in G, but do not span a fuzzy triangle in G. The Gallai fuzzy graphs are fuzzy spanning Gallai sub graphs of the well-known Class of fuzzy line graphs. Let γ(Γ(G)) and i(Γ(G)) denote the minimum fuzzy cardinality of a fuzzy dominating set of a Gallai fuzzy graph Γ(G)= (σ, μ) with n fuzzy vertices and maximum fuzzy degree Δ(Γ(G)), (Γ(G))≤ nΔ(Γ(G), i(G) ≤ n-Δ(Γ(G)). In this manuscript, we characterized the fuzzy connected bipartite Gallai fuzzy graphs which achieve this upper bound. Here, we have shown that an arbitrary Gallai fuzzy Graph G are furnished with two conditions which are necessary if γ(Γ(G)) + Δ(Γ(G))= n and are sufficient to achieve n 1 ≤ γ(Γ(G))+Δ(Γ(G)) ≤ n. Keywords— Gallai fuzzy graph, Gallai-type theorems, fuzzy domination parameters

This article analyzes F\olner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70ies. We prove that each essentially hyponormal operator has a proper F\olner sequence (i.e. a F\olner sequence of projections strongly converging to 1). In particular, any quasinormal, any subnormal, any hyponormal and any essentially normal operator has a proper F\olner sequence. Moreover, we show that an operator is finite if and only if it has a proper F\olner sequence or if it has a non-trivial finite dimensional reducing subspace. We also analyze the structure of operators which have no F\olner sequence and give examples of them. For this analysis we introduce the notion of strongly non-F\olner operators, which are far from finite block reducible operators, in some uniform sense, and show that this class coincides with the class of non-finite operators.

In this paper we discuss copositive tensors, which are a natural generalization of the copositive matrices. We present an analysis of some basic properties of copositive tensors; as well as the conditions under which class of copositive tensors and the class of positive semidefinite tensors coincides. Moreover, we have describe several hierarchies that approximates the cone of copositive tensors. The hierarchies are predominantly based on different regimes such as; simplicial partition, rational griding and polynomial conditions. The hierarchies approximates the copositive cone either from inside (inner approximation) or from outside (outer approximation). We will also discuss relationship among different hierarchies.