Degree Sequence Research Articles

Time Irreversibility of Complex Time Series with Detrended Cross-Correlation Analysis Based on Visibility Graph

In this paper, we introduce detrended cross-correlation analysis (DCCA) method into visibility graph (VG) algorithm and propose VGDCCA method to reflect the time series irreversibility from a new perspective. The validity and reliability of the proposed VGDCCA method is supported by numerical simulations on synthesized short-term correlated chaotic systems and long-term correlated fractal processes, and by the empirical analysis on stock indices and traffic parameter. The VGDCCA planes suggest that autoregressive fractionally integrated moving average (ARFIMA) and fractional Gaussian noise (FGN) series show time reversible behavior, whatever the long-term correlation of the series is or however strong the persistence or anti-persistence of the series is. Meanwhile, Logistic map with [Formula: see text] and Hénon map show more time irreversible behaviors than those of ARFIMA, FGN series and other Logistic maps. It can be found that the relationship between cross-correlation of ingoing degree sequence and outgoing degree sequence for the simulated series with different parameters is consistent with the complexity and autocorrelation behavior in the corresponding definition of the time series. For the empirical analysis, VGDCCA method declares the similarity and dissimilarity between stock indices on time series irreversibility and captures the time irreversibility of traffic time series recorded by the detectors in different locations.

Sampling random graphs with specified degree sequences

The configuration model is a standard tool for uniformly generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network’s structure can be explained by its degree structure alone. A Markov chain Monte Carlo (MCMC) algorithm, based on a degree-preserving double-edge swap, provides an asymptotic solution to sample from the configuration model. However, accurately and efficiently detecting when this Markov chain is sufficiently close to its stationary distribution remains an unsolved problem. Here, we provide a solution to sample from the configuration model using this standard MCMC algorithm. We develop an algorithm, based on the assortativity of the sampled graphs, for estimating the gap between effectively independent MCMC states, and a computationally efficient gap-estimation heuristic derived from analyzing a corpus of 509 empirical networks. We provide a convergence detection method based on the Dickey-Fuller Generalized Least Squares test, which we show is more accurate and efficient than three alternative Markov chain convergence tests. Supplementary materials for the proposed methods can be found here.

Embedding theorems for random graphs with specified degrees

Abstract Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$ , let ${\mathcal G}(n,W)$ be the random graph obtained by independently including each edge $jk\in \binom{[n]}{2}$ with probability $W_{jk}=W_{kj}$ . Given a degree sequence $\textbf{d}=(d_1,\ldots, d_n)$ , let ${\mathcal G}(n,\textbf{d})$ denote a uniformly random graph with degree sequence $\textbf{d}$ . We couple ${\mathcal G}(n,W)$ and ${\mathcal G}(n,\textbf{d})$ together so that asymptotically almost surely ${\mathcal G}(n,W)$ is a subgraph of ${\mathcal G}(n,\textbf{d})$ , where $W$ is some function of $\textbf{d}$ . Let $\Delta (\textbf{d})$ denote the maximum degree in $\textbf{d}$ . Our coupling result is optimal when $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$ , that is, $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for every $i,j\in [n]$ . We also have coupling results for $\textbf{d}$ that are not constrained by the condition $\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$ . For such $\textbf{d}$ our coupling result is still close to optimal, in the sense that $W_{ij}$ is asymptotic to ${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for most pairs $ij\in \binom{[n]}{2}$ .

The Eisenbud-Green-Harris conjecture for fast-growing degree sequences

Let $S$ be a standard graded polynomial ring over a field, and $I$ be a homogeneous ideal that contains a regular sequence of degrees $d_1,\ldots,d_n$. We prove the Eisenbud-Green-Harris conjecture when the forms of the regular sequence satisfy $d_i \geqslant \sum_{j=1}^{i-1}(d_j-1)$, improving a result obtained in 2008 by the first author and Maclagan. Except for the sporadic case of a regular sequence of five quadrics, recently proved by Gunturkun and Hochster, the results of this article recover all known cases of the conjecture where only the degrees of the regular sequence are fixed, and include several additional ones.

On a conjecture that strengthens Kundu's k $k$‐factor theorem

AbstractLet be a nonincreasing degree sequence with even . In 1974, Kundu showed that if is graphic, then some realization of has a ‐factor. For , Busch et al. and later Seacrest for showed that if and is graphic, then there is a realization with a ‐factor whose edges can be partitioned into a ‐factor and edge‐disjoint 1‐factors. We improve this to any . In 1978, Brualdi and then Busch et al. in 2012, conjectured that . The conjecture is still open for . However, Busch et al. showed the conjecture is true when or . We explore this conjecture by first developing new tools that generalize edge‐exchanges. With these new tools, we can drop the assumption is graphic and show that if then has a realization with edge‐disjoint 1‐factors. From this we confirm the conjecture when or when is graphic and .

Approximate realizations for outerplanaric degree sequences

We study the question of whether a sequence d=(d1,…,dn) of positive integers is the degree sequence of some outerplanar graph G. If so, G is an outerplanar realization of d and d is an outerplanaric sequence. The case where ∑d≤2n−2 is easy, as d has a realization by a forest. In this paper, we consider the family D of all sequences d of even sum 2n≤∑d≤4n−6−2ω1, where ωx is the number of x's in d. We partition D into two disjoint subfamilies, D=DNOP∪D2PBE, such that every sequence in DNOP is provably non-outerplanaric, and every sequence in D2PBE is given a realizing graph G enjoying a 2-page book embedding (and moreover, one of the pages is also bipartite).

ALGEBRAIC MULTIPLICITY OF EIGENVALUE ZERO OF CERTAIN BI-REGULAR GRAPHS

A bi-regular graph is a graph whose degree sequence is exactly two. In this article, we give the algebraic multiplicity of the eigenvalue zero of certain bi-regular graphs.

An algebraic approach to the reconstruction of uniform hypergraphs from their degree sequence

The reconstruction of a (hyper)graph starting from its degree sequence is one of the most relevant among the inverse problems investigated in the field of graph theory. In case of graphs, a feasible solution can be quickly reached, while in case of hypergraphs Deza et al. (2018) proved that the problem is NP-hard even in the simple case of 3-uniform ones. This result opened a new research line consisting in the detection of instances for which a solution can be computed in polynomial time. In this work we deal with 3-uniform hypergraphs, and we study them from a different perspective, exploiting a connection of these objects with partially ordered sets. More precisely, we introduce a simple partially ordered set, whose ideals are in bijection with a subclass of 3-uniform hypergraphs. We completely characterize their degree sequences in case of principal ideals (here a simple fast reconstruction strategy follows), and we furthermore carry on a complete analysis of those degree sequences related to the ideals with two generators. We also consider unique hypergraphs in Dext, i.e., those hypergraphs that do not share their degree sequence with other non-isomorphic ones. We show that uniqueness holds in case of hypergraphs associated to principal ideals, and we provide some examples of hypergraphs in Dext where this property is lost.

Chromatic symmetric functions and polynomial invariants of trees

AbstractStanley asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach Stanley's problem by studying the relationship between the chromatic symmetric function and other invariants. First, we prove Crew's conjecture that the chromatic symmetric function of a tree determines its generalized degree sequence, which enumerates vertex subsets by cardinality and the numbers of internal and external edges. Second, we prove that the restriction of the generalized degree sequence to subtrees contains exactly the same information as the subtree polynomial, which enumerates subtrees by cardinality and number of leaves. Third, we construct arbitrarily large families of trees sharing the same subtree polynomial, proving and generalizing a conjecture of Eisenstat and Gordon.

Large induced subgraphs of random graphs with given degree sequences

AbstractWe study a random graph with given degree sequence , with the aim of characterising the degree sequence of the subgraph induced on a given set of vertices. For suitable and , we show that the degree sequence of the subgraph induced on is essentially concentrated around a sequence that we can deterministically describe in terms of and . We then give an application of this result, determining a threshold for when this induced subgraph contains a giant component. We also apply a similar analysis to the case where is chosen by randomly sampling vertices with some probability , that is, site percolation, and determine a threshold for the existence of a giant component in this model. We consider the case where the density of the subgraph is either constant or slowly going to 0 as goes to infinity, and the degree sequence of the whole graph satisfies a certain maximum degree condition. Analogously, in the percolation model we consider the cases where either is a constant or where slowly. This is similar to work of Fountoulakis in 2007 and Janson in 2009, but we work directly in the random graph model to avoid the limitations of the configuration model that they used.

Graphs with degree sequence [formula omitted] and [formula omitted]

In this paper we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. We establish various properties of Gm,n and G¯m,n related to recognition, connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We also show that several classical optimization problems on these graphs are NP-hard, while others can be solved efficiently.

Sequential Stub Matching for Asymptotically Uniform Generation of Directed Graphs with a Given Degree Sequence

AbstractWe discuss sequential stub matching for directed graphs and show that this process can be used to sample simple digraphs with asymptotically equal probability. The process starts with an empty edge set and repeatedly adds edges to it with a certain state-dependent bias until the desired degree sequence is fulfilled, whilst avoiding placing a double edge or self-loop. We show that uniform sampling is achieved in the sparse regime when the maximum degree $$d_\text {max}$$ d max is asymptotically dominated by $$m^{1/4}$$ m 1 / 4 , where m is the number of edges. The proof is based on deriving various combinatorial estimates related to the number of digraphs with a given degree sequence and controlling concentration of these estimates in large digraphs. This suggests that sequential stub matching can be viewed as a practical algorithm for almost uniform sampling of digraphs. We show that this algorithm can be implemented to feature a linear expected runtime O(m).

Mathematical modeling for prediction of physicochemical characteristics of cardiovascular drugs via modified reverse degree topological indices.

Global health concerns persist due to the multifaceted nature of heart diseases, which include lifestyle choices, genetic predispositions, and emerging post-COVID complications like myocarditis and pericarditis. This broadens the spectrum of cardiovascular ailments to encompass conditions such as coronary artery disease, heart failure, arrhythmias, and valvular disorders. Timely interventions, including lifestyle modifications and regular medications such as antiplatelets, beta-blockers, angiotensin-converting enzyme inhibitors, antiarrhythmics, and vasodilators, are pivotal in managing these conditions. In drug development, topological indices play a critical role, offering cost-effective computational and predictive tools. This study explores modified reverse degree topological indices, highlighting their adjustable parameters that actively shape the degree sequences of molecular drugs. This feature makes the approach suitable for datasets with unique physicochemical properties, distinguishing it from traditional methods that rely on fixed degree approaches. In our investigation, we examine a dataset of 30 drug compounds, including sotagliflozin, dapagliflozin, dobutamine, etc., which are used in the treatment of cardiovascular diseases. Through the structural analysis, we utilize modified reverse degree indices to develop quantitative structure-property relationship (QSPR) models, aiming to unveil essential understandings of their characteristics for drug development. Furthermore, we compare our QSPR models against the degree-based models, clearly demonstrating the superior effectiveness inherent in our proposed method.

Predicting graph energy and entropy analysis of pent-heptagonal nanomaterials: Insights from regression models using generalized reverse degree-sum topological indices

Nanostructures have sparked notable interest in materials science due to their diverse applications, especially carbon nanosheets with pentagonal–heptagonal variations have emerged as subjects of intense scientific investigation. These nanostructures show great potential in a wide range of fields, including energy storage, optoelectronics, catalysis, and bioelectronics, thereby making substantial contributions to nanotechnology and material science. Topological indices hold a pivotal position in the domain of mathematical and computational chemistry providing numerical information about molecular structures. This information is instrumental in establishing predictive relationships with chemical and physical properties, finding applications across various fields such as drug development and materials science. This paper introduces a novel generalized version of reverse degree-sum based topological indices that effectively shaping the degree sequence of nanomaterials to fit the physical and chemical properties of dataset, differentiating it from conventional fixed-degree methods. Consequently, we explore entropy-based methodologies to compare the structural complexities of pent-heptagonal nanosheets. Furthermore, our study predicts the graph energies of these nanosheets via linear regression equations, comparing them with energy values predicted by the McClelland equation.

Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations

Let $G = (V(G),E(G))$ be a graph with degree sequence $\langle d_1,d_2, \cdots, d_n \rangle$, where $d_1 \ge d_2 \ge \cdots \ge d_n$. The polynomial representation of $G$ is given by $f_G(x) = \displaystyle \sum_{i=1}^n x^{d_i} = \sum_{k=1}^{\Delta(G)}a_kx^{k}$, where $a_k$ is the number of vertices of $G$ having degree $k$ for each $i = 1,2,\cdots n = \Delta(G)$. In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs.

Asymptotic in a class of network models with an increasing sub-Gamma degree sequence

. For differential privacy under sub-Gamma noise, we derive the asymptotic properties of a class of network models with binary values with a general link function. In this article, we release the degree sequences of the binary networks under a general noisy mechanism, with the discrete Laplace mechanism as a special case. We establish the asymptotic result, including both consistency and asymptotically normality, of the parameter estimator when the number of parameters goes to infinity in a class of network models. Simulations and a real-data example are provided to illustrate the asymptotic results.

The number of 1-nearly independent vertex subsets

Let G be a graph with vertex set V(G) and edge set E(G). A subset I of V(G) is an independent vertex subset if no two vertices in I are adjacent in G. We study the number, σ 1(G), of all subsets of V(G) that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of σ 1 are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on σ 1 for graphs of order n. We deduce as a corollary that the star K 1,n−1 (the tree with degree sequence (n − 1, 1, . , 1)) is the n-vertex tree with smallest σ 1, while it is well known that K1,n−1 is the n-vertex tree with largest number of independent subsets.

Behavior and shear strength of prestressed steel reinforced concrete composite deep beam

In order to optimise the utilisation of material properties, simplify the construction process and enhance the industrialisation of steel reinforced concrete beams, an innovative prestressed steel reinforced concrete composite beams(PSRCC) was proposed, which was achieved by integrating prestressing technology and prefabricated technology to steel reinforced concrete structure. To investigate the shear PSRCC deep beams, a static load test was conducted on 7 PSRCC deep beams. The study investigated the effects of parameters such as shear span-depth ratio, prestressed tendons degree, form, height and application sequence on the failure mode, crack pattern, load-displacement behavior, strain response and damage evolution. A model was established to predict the shear strength of PSRCC deep beams, considering the steel-concrete interaction, using the von Mises yield criterion and the SST model, which assessed the contributions of composite strut, prestressed tendons and shear steel web for shear strength. The test results showed that the prefabricated and cast-in-situ concrete components maintained good integrity, and PSRCC deep beams eventually experienced typical diagonal compression failure. Applying prestressing and reducing the shear span-depth ratio under identical conditions would reduce the deformation capacity of PSRCC deep beams. When the shear span-depth ratio was reduced from 1.0 to 0.5, the cracking load and peak load of the specimens increased by 17.31 % and 16.53 % respectively. Increasing the prestress level from 0 to 0.3 and 0.6 increased the cracking load of the specimens by 1.41 and 2.36 times, respectively, but the peak load of the specimens increased by only 3.37 % and 8.17 %. The cracking load of the parabolic tendon specimen increased by 2.18 for the comparison specimen but was 8.33 % lower than that of the linear tendon specimen with the same degree of prestressing. Comparison of the two prestressing sequences showed that the cracking load of the specimen with tensioning prestressed tendons after pouring the entire beam section increased by 29.84 %, but the peak load increased by only 3.11 %. As the height of the prestres tendons increased, the cracking and peak loads of the specimen reduced by 5.84 % and 5.71 % respectively. Consequently, the shear strength of PSRCC deep beams was significantly influenced by the shear span-depth ratio and minimally affected by prestress-related parameters. The proposed model accurately predicted the shear strength of PSRCC deep beams compared to existing codes, offering valuable guidance for design.

Rao's Theorem for forcibly planar sequences revisited

We consider the graph degree sequences such that every realisation is a polyhedron. It turns out that there are exactly eight of them. All of these are unigraphic, in the sense that each is realised by exactly one polyhedron. This is a revisitation of a Theorem of Rao about sequences that are realised by only planar graphs.Our proof yields additional geometrical insight on this problem. Moreover, our proof is constructive: for each graph degree sequence that is not forcibly polyhedral, we construct a non-polyhedral realisation.

Structural properties of extended pseudo-fractal scale-free network with higher network efficiency

Abstract Complex networks, as an important model for studying complex systems, have been gradually studied in various extension models. Firstly, the network is proved to be sparse by calculating the density of the extending pseudo-fractal network. Secondly, it is proved that the network nodes conform the power-law distribution by enumerate the degree sequence of nodes with power law index $ 2 \lt \gamma \leq 2.585 $ shows that the network has scale-free feature. Thirdly, it is also demonstrated that the average clustering coefficient $ \overline{C_{g}} $ can reach a theoretical lower limit value 0.8182 and the increase of extension coefficient m makes the $ \overline{C_{g}} $ higher than traditional fractal networks. Fourthly, we derive the analytical expression and limit expression of the average path length, and conclude that it has small-world effect. Meanwhile, it is shown that the average path length $ \overline{D_{g}} $ is logarithmically related to Ng growth relationship in Fg. Finally, it is concluded that this extended pseudo-fractal network becomes more effective under the effect of m.