Degree Sequence Research Articles

The Signless p-Laplacian Spectral Radius of Graphs with Given Degree Sequences

The Signless p-Laplacian Spectral Radius of Graphs with Given Degree Sequences

Synthesis of contracted graph for planar nonfractionated simple-jointed kinematic chain based on similarity information

Many structural synthesis methods require a complete database of contracted graphs. Therefore, this study proposes a method based on similarity information to efficiently synthesize contracted graphs for planar nonfractionated simple-jointed kinematic chains. First, the degree of all vertices is calculated using the number of independent loops and the degree sequence equation. Then, the method of vertex connection is used to enumerate the contracted graph. In this process, similarity information is used to match the connection type between vertices quickly. The type of similar vertex is determined, and nonisomorphic graphs caused by incompletely equivalent similar vertices are discovered. Additionally, the changes in similarity information are tracked to identify nonisomorphic graphs caused by similarity variation. Then, the characteristic constants representing the similarity of graphs are retained to rule out the isomorphism caused by different parent graphs. Finally, the rationality of contracted graphs is identified. The contracted graph with up to eight independent loops and its solution time are provided. Furthermore, the reliability and efficiency of the method can be demonstrated by comparing it to the database of other literature.

Understanding the relevance of experiential learning for entrepreneurial self-efficacy: A gender-wise perspective

Experiential learning has been regarded as an effective tool for grasping entrepreneurial knowledge. The current study examines the influence of Experiential Learning (EL) with sub-variables [(i.e., Concrete Experience (CE), Reflective Observation (RO), Abstract Conceptualisation (AC), & Active Experimentation (AE)] on the Entrepreneurial self-efficacy (ESE) of the male and female Higher educational institution (HEI) students possessing high Entrepreneurial Intention. The study uses a Liñán and Chen (2009) tool to identify students with high Entrepreneurial Intentions (EI), selecting 323 (195 male and 128 female) respondents possessing 4–5 scores on a scale of 5. The survey considered three growing Indian regions: Punjab, Haryana, and NCR (National capital region). For EL, a self-structured questionnaire and for the ESE scale of Mcgee, Peterson, Mueller, and Sequeira (2009) was adopted. The results highlight that in the case of male and female students, all sub-variables of EL, i.e., CE, RO, AC & AE of EL, strongly predicted ESE. However, there is a difference in the degree of influence of such dimensions based on Gender. In the case of male degree of influence is maximum with RO followed by AE, CE, AC, but for females the sequence of degree of influence is different as RO is followed by CE, AC and AE. The findings can be helpful to design and support experiential learning system concerning entrepreneurship students of higher educational institutes.

Parallel global edge switching for the uniform sampling of simple graphs with prescribed degrees

The uniform sampling of simple graphs matching a prescribed degree sequence is an important tool in network science, e.g. to construct graph generators or null-models. Here, the Edge Switching Markov Chain (ES-MC) is a common choice. Given an arbitrary simple graph with the required degree sequence, ES-MC carries out a large number of small changes, called edge switches, to eventually obtain a uniform sample. In practice, reasonably short runs efficiently yield approximate uniform samples.In this work, we study the problem of executing edge switches in parallel. We discuss parallelizations of ES-MC, but find that this approach suffers from complex dependencies between edge switches. For this reason, we propose the Global Edge Switching Markov Chain (G-ES-MC), an ES-MC variant with simpler dependencies.We show that G-ES-MC converges to the uniform distribution and design shared-memory parallel algorithms for ES-MC and G-ES-MC. In an empirical evaluation, we provide evidence that G-ES-MC requires not more switches than ES-MC (and often fewer), and demonstrate the efficiency and scalability of our parallel G-ES-MC implementation.

Largest component of subcritical random graphs with given degree sequence

We study the size of the largest component of two models of random graphs with prescribed degree sequence, the configuration model (CM) and the uniform model (UM), in the (barely) subcritical regime. For the CM, we give upper bounds that are asymptotically tight for certain degree sequences. These bounds hold under mild conditions on the sequence and improve previous results of Hatami and Molloy on the barely subcritical regime. For the UM, we give weaker upper bounds that are tight up to logarithmic terms but require no assumptions on the degree sequence. In particular, the latter result applies to degree sequences with infinite variance in the subcritical regime.

The degree sequence on tensor and cartesian products of graphs and their omega index

<abstract><p>The aim of this paper is to illustrate how degree sequences may successfully be used over some graph products. Moreover, by taking into account the degree sequence, we will expose some new distinguishing results on special graph products. We will first consider the degree sequences of tensor and cartesian products of graphs and will obtain the omega invariant of them. After that we will conclude that the set of graphs forms an abelian semigroup in the case of tensor product whereas this same set is actually an abelian monoid in the case of cartesian product. As a consequence of these two operations, we also give a result on distributive law which would be important for future studies.</p></abstract>

Chebyshev's Sum Inequality and the Zagreb Indices Inequality

In a recent article, Nadeem and Siddique used Chebyshev’s sum inequality to establish the Zagreb indices inequality M1/n ≤ M2/m for undirected graphs in the case where the degree sequence (di) and the degree-sum sequence (Si) are similarly ordered. We show that this is actually not a completely new result and we discuss several related results that also cover similar inequalities for directed graphs, as well as sum-symmetric matrices and Eulerian directed graphs.

A Note on Extremal Sombor Indices of Trees with a Given Degree Sequence

We note here that the problem of determining extremal values of Sombor index for trees with a given degree sequence fits within the framework of results by Hua Wang from [Cent. Eur. J. Math. 12 (2014) 1656-1663], implying that the greedy tree has the minimum Sombor index, while an alternating greedy tree has the maximum Sombor index.

Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture

Abstract An n-vertex graph is called C-Ramsey if it has no clique or independent set of size $C\log _2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.

Bounds for the reverse (delta) first Zagreb indices

Let G = (V, E) be a simple connected graph with n ≥ 2 vertices and m edges. In the literature four vertex–degree–based topological indices are associated to topological index, known as the first Zagreb index, based on the modification of a sequence of degrees. In this paper we study relationships between these topological indices and establish some new bounds.

An alternative proof of the Sombor index minimizing property of greedy trees

Recently, Gutman defined a new graph invariant which is named the Sombor index SO(G) of a graph G and is computed via the expression SO(G) = ?u~v? qdeg(u)2 + deg(v)2, where deg(u) represents the degree of the vertex u in G and the summing is performed across all the unordered pairs of adjacent vertices u and v. Damnjanovic et al. have implemented an earlier result obtained by Wang in order to show that, among all the trees TD that have a specified degree sequence D, the greedy tree must attain the minimum Sombor index. Here we provide an alternative proof of this same result by constructing an auxiliary graph invariant named the pseudo-Sombor index and without relying on any other earlier results.

Remark on delta and reverse inverse degree indices

Let G = (V, E), V = {v1, v2, . . . , vn}, be a simple graph of order n and size m, with vertex–degree sequence ∆ = d1 ≥ d2 ≥ · · · ≥ dn = δ > 0. and let s1 ≥ s2 ≥ · · · ≥ sn, si = di − δ + 1 and c1 ≤ c2 ≤ · · · ≤ cn, ci = ∆ − di + 1, be two vertex–degree like sequences. By analogy with the inverse degree graph invariant, ID(G) = ∑n i=1 1 di , the delta inverse degree and reverse inverse degree indices are defined, respectively, as δID(G) = ∑n i=1 1 si and RID(G) = ∑n i=1 1 ci . In this paper we determine sharp bounds on δID(G) and RID(G) and the extremal graphs are characterized.

The Foata–Fuchs proof of Cayley’s formula, and its probabilistic uses

We present a very simple bijective proof of Cayley’s formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.

A General Computational Approach for Counting Labeled Graphs

This paper presents a general recursive formula to estimate the number of labeled graphs as well as details to evaluate the formula for the following graph properties: number of edges (graph density), degree sequence, degree distribution, classification mixing, and degree mixing, i.e., the formula estimates the number of labeled graphs that have given values for graph properties. The proposed approach can be extended to additional graph properties (e.g., number of triangles) as well as properties of bipartite graphs. For special settings in which formulas exist from previous research, simulation studies demonstrate the validity of the proposed approach. In addition, we demonstrate how our approach can be used to quantify the level of variability in values of a graph property in the subset of graphs that hold a specified value of a different graph property (or properties) constant.

Inverse Sum Indeg Index (Energy) with Applications to Anticancer Drugs

For a simple graph with vertex set {v1,v2,…,vn} with degree sequence dvi of vertex vi,i=1,2,…,n, the inverse sum indeg matrix (ISI-matrix) AISI(G)=(aij)n×n of G is defined by aij=dvidvjdvi+dvj, if vi is adjacent to vj, and zero, otherwise. The multiset of eigenvalues of AISI(G) is the ISI-spectrum of G and the sum of their absolute values is the ISI-energy of G. In this paper, we modify the two results of (Li, Ye and Broersma, 2022), give the correct characterization of the extremal graphs and thereby obtain better bounds than the already known results. Moreover, we also discuss the QSPR analysis and carry the statistical modelling (linear, logarithmic and quadratic) of the physicochemical properties of anticancer drugs with the ISI-index (energy).

Алгоритм розкладу цілих чисел і гладкого наближення функцій

The problem of expansion in powers is generalized into decomposition of positive integers in the sequence of degrees of different orders, the con-ditions of decomposition are determined, and the algorithm for decomposi-tion is constructed. The algorithm is based on two procedures: 1) achieve-ment a minimum of residual at each algorithm step; 2) speeding of decom-position through expanding the local base by reducing decomposition in-dex, which ensures finiteness of algorithm. The algorithm has such effi-ciency factors as high rate of decomposition, ease of implementation, availability of different options for the decomposition of numbers as in ex-tended, narrowed, sparse bases, which protects the encoded information from external influences. The algorithm can be used to encode large amounts of digital information under basic systems of small dimensions. Decomposition of positive integers into a sequence of powers is opti-mal and correct. Optimality of decomposition follows from the condition that at each step of algorithm the minimum value of disjunction in the space of mixed parameters x∈N,y∈Ris achieved. Correctness of algo-rithm is due to the fact that when the disjunction is reduced, the algorithm expands the basis of decomposition by reducing the degree indicators by one. By switching from a discrete model to a continuous model by replac-ing the degrees with power functions, we obtain a smooth approximation of the ill-conditioned function in the neighborhood of decomposition. The construction of posinomial polynomials on the basis of smooth polynomi-als is one of the promising directions of integration of ill-conditioned non-differentiable functions and smooth replacement of variables in the catas-trophe theory.Posinomials (functions with a variable exponent) predict the step of splitting the integration interval into parts, since they determine the loga-rithmic rate of change of an arbitrary monotonic function. The method of decomposition of positive integers provides an optimal decomposition into the sum of powers, and therefore the transition from a discrete model to a continuous model in the neighborhood of decomposition by replacing powers with power functions as well as allows to achieve the high accura-cy of approximation.

Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs

Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as L E = ∑ i = 1 n | μ i - 2 m n | LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|} , L E L = ∑ i = 1 n - 1 μ i LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} } and K f = n ∑ i = 1 n - 1 1 μ i Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}} respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.

Reconstruction of helm graph and web graph

The graph reconstruction conjecture asserts that every simple undirected graph with number of vertices (where <em>n</em> ≥ 3) is uniquely determined up to isomorphism, by its collection of unlabeled form of vertex deleted sub graphs. If a vertex deleted sub graph of <em>G</em> is given in unlabeled form, then it is called a “card” of <em>G</em>. The collection of all cards of <em>G</em> is called the “deck” of <em>G</em>, and is denoted by <em>D</em>(<em>G</em>) . If we can determine the original graph <em>G</em> from the deck of the graph <em>G</em>, we can say that it is reconstructible. In this paper we propose a new method to reconstruct the helm graph (<em>H_n</em>) and the web graph (<em>W_n</em>), by using the degree sequence of its collection of vertex deleted sub graphs. Helm graph is obtained by adjoining a pendant edge to each node of the cycle of the -wheel graph. Web graph is obtained by joining pendant vertices to the vertices in the outer cycle of the helm graph to form a cycle and again adding pendant vertices to the new cycle. For both graphs we found decks and obtained the degree sequence of each cards. Finally, we reconstructed the original graph using the degree sequences.

Fundamental Invariants of Tensors, Latin Hypercubes, and Rectangular Kronecker Coefficients

Abstract We study polynomial SL-invariants of tensors, mainly focusing on fundamental invariants that are of smallest degrees. In particular, we prove that certain 3-dimensional analogue of the Alon–Tarsi conjecture on Latin cubes considered previously by Bürgisser and Ikenmeyer implies positivity of (generalized) Kronecker coefficients at rectangular partitions and as a result provides values for degree sequences of fundamental invariants.

The Burge correspondence and crystal graphs

The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the simple graphs of hook shape by peak and valley conditions on Burge arrays. This is the first step towards an analogue of Schensted’s result for the RSK insertion which states that the length of the longest increasing subword of a word is the length of the largest row of the tableau under the RSK correspondence. Furthermore, we give a crystal structure on simple graphs of hook shape. The extremal vectors in this crystal are precisely the simple graphs whose degree sequence are threshold and hook-shaped.