New Polynomial-Based Molecular Descriptors with Low Degeneracy

In this paper, we introduce a novel graph polynomial called the ‘information polynomial’ of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power

Schultz and Modified Schultz Polynomials for Edge – Identification Chain and Ring – for Square Graphs

في البيان المتصل G، دالة المسافة بين أي رأسين من رؤوس البيان  هي أقصر مسافة بينهما، كما تعرف درجة الرأس  والتي يرمز لها بـ  بأنها عدد الحافات الواقعة عليه. متعددة حدود شوالتز وشوالتز المعدلة تعرف كالاتي:   على التوالي، حيث أن المجموع يؤخذ لكل الازواج غير المرتبة من الرؤوس المختلفة في  وأن  هي المسافة بين الرأسين   و  في . في هذا البحث استطعنا الحصول على صيغ عامة لكل من متعددة حدود شوالتز وشوالتز المعدلة ودليليهما ومعدلهما لتطابق الحافة لسلسلة وحلقة للبيانات المربعة.  In a connected graph , the distance function between each pair of two vertices from a set vertex  is the shortest distance between them and the vertex degree  denoted by  is the number of edges which are incident to the vertex  The Schultz and modified Schultz polynomials of  are have defined as:  respectively, where the summations are taken over all unordered pairs of distinct vertices in  and  is the distance between  and  in  The general forms of Schultz and modified Schultz polynomials shall be found and indices of the edge – identification chain and ring – square graphs in the present work

The Expected Values of Hosoya Index and Merrifield-Simmons Index of Random Hexagonal Cacti

Hosoya index and Merrifield-Simmons index are two well-known topological descriptors that reflex some physical properties, boiling point or heat of formation for instance, of bezenoid hydrocarbon compounds. In this paper, we establish the generating functions of the expected values of these two indices of random hexagonal cacti. This generalizes the results of Doslic and Maloy, published in Discrete Mathemaics, in 2010. By applying the ideas on meromorphic functions and the growth of power series coefficients, the asymptotic behaviors of these indices on the random cacti have been established.Comment: 27 pages, 4 figures, author's name spelling in references revise

Orbit Entropy and Symmetry Index Revisited

The size of the orbits or similar vertices of a network provides important information regarding each individual component of the network. In this paper, we investigate the entropy or information content and the symmetry index for several classes of graphs and compare the values of this measure with that of the symmetry index of certain graphs.publishedVersionPeer reviewe

On the relationship between PageRank and automorphisms of a graph

PageRank is an algorithm used in Internet search to score the importance of web pages. The aim of this paper is demonstrate some new results concerning the relationship between the concept of PageRank and automorphisms of a graph. In particular, we show that if vertices u and v are similar in a graph G (i.e., there is an automorphism mapping u to v), then u and v have the same PageRank score. More generally, we prove that if the PageRanks of all vertices in G are distinct, then the automorphism group of G consists of the identity alone. Finally, the PageRank entropy measure of several kinds of real-world networks and all trees of orders 10–13 and 22 is investigated.acceptedVersionPeer reviewe

Mapping Information Flow in Sensorimotor Networks

Biological organisms continuously select and sample information used by their neural structures for perception and action, and for creating coherent cognitive states guiding their autonomous behavior. Information processing, however, is not solely an internal function of the nervous system. Here we show, instead, how sensorimotor interaction and body morphology can induce statistical regularities and information structure in sensory inputs and within the neural control architecture, and how the flow of information between sensors, neural units, and effectors is actively shaped by the interaction with the environment. We analyze sensory and motor data collected from real and simulated robots and reveal the presence of information structure and directed information flow induced by dynamically coupled sensorimotor activity, including effects of motor outputs on sensory inputs. We find that information structure and information flow in sensorimotor networks (a) is spatially and temporally specific; (b) can be affected by learning, and (c) can be affected by changes in body morphology. Our results suggest a fundamental link between physical embeddedness and information, highlighting the effects of embodied interactions on internal (neural) information processing, and illuminating the role of various system components on the generation of behavior

Minimum-entropy causal inference and its application in brain network analysis

Identification of the causal relationship between multivariate time series is a ubiquitous problem in data science. Granger causality measure (GCM) and conditional Granger causality measure (cGCM) are widely used statistical methods for causal inference and effective connectivity analysis in neuroimaging research. Both GCM and cGCM have frequency-domain formulations that are developed based on a heuristic algorithm for matrix decompositions. The goal of this work is to generalize GCM and cGCM measures and their frequency-domain formulations by using a theoretic framework for minimum entropy (ME) estimation. The proposed ME-estimation method extends the classical theory of minimum mean squared error (MMSE) estimation for stochastic processes. It provides three formulations of cGCM that include Geweke's original time-domain cGCM as a special case. But all three frequency-domain formulations of cGCM are different from previous methods. Experimental results based on simulations have shown that one of the proposed frequency-domain cGCM has enhanced sensitivity and specificity in detecting network connections compared to other methods. In an example based on in vivo functional magnetic resonance imaging, the proposed frequency-domain measure cGCM can significantly enhance the consistency between the structural and effective connectivity of human brain networks

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Biological Network Analysis and Comparison

Ph.DDOCTOR OF PHILOSOPH

Spacetime geometry from graviton condensation

In this thesis we introduce a novel approach viewing spacetime geometry as an emergent phenomenon based on the condensation of a large number of quanta on a distinguished flat background. We advertise this idea with regard to investigations of spacetime singularities within a quantum field theoretical framework and semiclassical considerations of black holes. Given that in any physical theory apart from General Relativity the metric background is determined in advance, singularities are only associated with observables and can either be removed by renormalization techniques or are otherwise regarded as unphysical. The appearance of singularities in the spacetime structure itself, however, is pathological. The prediction of said singularities in the sense of geodesic incompleteness culminated in the famous singularity theorems established by Hawking and Penrose. Though these theorems are based on rather general assumptions we argue their physical relevance. Using the example of a black hole we show that any classical detector theory breaks down far before geodesic incompleteness can set in. Apart from that, we point out that the employment of point particles as diagnostic tools for spacetime anomalies is an oversimplification that is no longer valid in high curvature regimes. In view of these results the question arises to what extent quantum objects are affected by spacetime singularities. Based on the definition of geodesic incompleteness customized for quantum mechanical test particles we collect ideas for completeness concepts in dynamical spacetimes. As it turns out, a further development of these ideas has shown that Schwarzschild black holes, in particular, allow for a evolution of quantum probes that is well-defined all over. This fact, however, must not distract from such semiclassical considerations being accompanied by many so far unresolved paradoxes. We are therefore compelled to take steps towards a full quantum resolution of geometrical backgrounds. First steps towards such a microscopic description are made by means of a non-relativistic scalar toy model mimicking properties of General Relativity. In particular, we model black holes as quantum bound states of a large number N of soft quanta subject to a strong collective potential. Operating at the verge of a quantum phase transition perturbation theory naturally breaks down and a numerical analysis of the model becomes inevitable. Though indicating 1/N corrections as advertised in the underlying so-called Quantum-N portrait relevant for a possible purification of Hawking radiation and henceforth a resolution of the long-standing information paradox we recognize that such a non-relativistic model is simply not capable of capturing all relevant requirements of a proper black hole treatment. We therefore seek a relativistic framework mapping spacetime geometry to large-N quantum bound states. Given a non-trivial vacuum structure supporting graviton condensation this is achieved via in-medium modifications that can be linked to a collective binding potential. Viewing Minkowski spacetime as fundamental, the classical notion of any other spacetime geometry is recovered in the limit of an infinite constituent number of the corresponding bound state living on Minkowski. This construction works in analogy to the description of hadrons in quantum chromodynamics and, in particular, also uses non-perturbative methods like the auxiliary current description and the operator product expansion. Concentrating on black holes we develop a bound state description in accordance with the isometries of Schwarzschild spacetime. Subsequently, expressions for the constituent number density and the energy density are reviewed. With their help, it can be concluded that the mass of a black hole at parton level is proportional to its constituent number. Going beyond this level we then consider the scattering of a massless scalar particle off a black hole. Using previous results we can explicitly show that the constituent distribution represents an observable and therefore might ultimately be measured in experiments to confirm our approach. We furthermore suggest how the formation of black holes or Hawking radiation can be understood within this framework. After all, the gained insights, capable of depriving their mysteries, highlights the dubiety of treating black holes by means of classical tools. Since our setup allows to view other, non-black-hole geometries, as bound states as well, we point out that our formalism could also shed light on the cosmological constant problem by computing the vacuum energy in a de Sitter state. In addition, we point our that even quantum chromodynamics, and, in fact, any theory comprising bound states, can profit from our formalism. Apart from this, we discuss an alternative proposal describing classical solutions in terms of coherent states in the limit of an infinite occupation number of so-called corpuscles. Here, we will focus on the coherent state description of Anti-de Sitter spacetime. Since most parts of this thesis are directed to find a constituent description of black holes we will exclude this corpuscular description from the main part and introduce it in the appendix