Minimum Degrees Research Articles

Quantum walks driven by quantum coins with two multiple eigenvalues

We consider a spectral analysis on the quantum walks on graph $$G=(V,E)$$ with the local coin operators $$\{C_u\}_{u\in V}$$ and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues $$\kappa ,\kappa '$$ and $$p=\dim (\ker (\kappa -C_u))$$ for any $$u\in V$$ with $$1\le p\le \delta (G)$$ , where $$\delta (G)$$ is the minimum degrees of G. We show that this quantum walk can be decomposed into a cellular automaton on $$\ell ^2(V;\mathbb {C}^p)$$ whose time evolution is described by a self adjoint operator T and its remainder. We obtain how the eigenvalues and its eigenspace of T are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on $$\mathbb {Z}^d$$ with the moving shift in the Fourier space.

Thermomechanical Behaviors of an Energy Pile–Raft Foundation under Intermittent Operation with Forced Heat Recharge

This paper presents an experimental investigation of the thermal and mechanical behaviors of an energy pile–raft foundation under intermittent heating operation with forced heat recharge, with 16 h of pile cooling and 8 h of forced heat recharge (referred to as 16F, where F indicates forced). The forced heat recharge was found to increase the daily heat exchange rate to 180 W/m, which was 50% higher than the value under general intermittent heating operation without forced heat recharge. The raft and building load at the pile head imposed a significant restraint on the upper part of the energy pile, leading to a minimum degrees of freedom (0.27) near the pile head. The mechanical behaviors of the pile, including strains, stresses, and pile end displacements and stresses, followed reversible paths when plotted against the daily temperature cycles. A slight residual pressure stress (10 kPa) was observed at the bottom of the raft during the initial thermal cycles.

On the outer-independent double Italian domination number

‎ An outer-independent Italian dominating function (OIIDF) on a graph G is a function f : V ( G )→{0, 1, 2} such that every vertex v ∈ V ( G ) with f ( v )=0 has at least two neighbors assigned 1 under f or one neighbor w with f ( w )=2 , and the set { u ∈ V ( G )| f ( u )=0} is independent. An outer-independent double Italian dominating function (OIDIDF) on a graph G is a function f : V ( G )→{0, 1, 2, 3} such that if f ( v )∈{0, 1} for a vertex v ∈ V ( G ) , then ∑ u ∈ N [ v ] f ( u )≥3 and the set { u ∈ V ( G )| f ( u )=0} is independent. The weight of an OIIDF (respectively, OIDIDF) f is the value w ( f )=∑ v ∈ V ( G ) f ( v ) . The minimum weight of an OIIDF (respectively, OIDIDF) on a graph G is called the outer-independent Italian (respectively, outer-independent double Italian) domination number of G . We characterize all trees T with outer-independent double Italian domination number twice the outer-independent Italian domination number. We also present lower bounds on the outer-independent double Italian domination number of a connected graph G in terms of the order, minimum and maximum degrees.

A distributed counting architecture for exploring the structure of anonymous active–passive networks

Active–passive networks are composed of nodes that meet specific criteria depending on applications (active nodes) and nodes that do not (passive nodes). This paper investigates the network size estimation problem, which involves determining the number of active nodes in anonymous active–passive networks in a distributed way. To this end, a distributed counting architecture is proposed to locally manipulate the criteria for being an active node via a variable and to count the number of active nodes, where this is accomplished without prior knowledge of the network or a procedure for initialization. Several properties of an anonymous network such as network size, number of nodes for a given degree, the maximum and minimum degrees, and so on, can be deduced using the proposed architecture by varying the criteria variable. Stability and convergence analysis of the proposed architecture are presented, and its effectiveness is also demonstrated through a numerical example.

The upper bounds on the Steiner k-Wiener index in terms of minimum and maximum degrees

The upper bounds on the Steiner k-Wiener index in terms of minimum and maximum degrees

Complete subgraphs in a multipartite graph

AbstractIn 1975 Bollobás, Erdős, and Szemerédi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$ , what is the largest minimum degree $\delta (G)$ among all $r$ -partite graphs $G$ with parts of size $n$ and which do not contain a copy of $K_{t+1}$ ? The $r=t+1$ case has attracted a lot of attention and was fully resolved by Haxell and Szabó, and Szabó and Tardos in 2006. In this article, we investigate the $r\gt t+1$ case of the problem, which has remained dormant for over 40 years. We resolve the problem exactly in the case when $r \equiv -1 \pmod{t}$ , and up to an additive constant for many other cases, including when $r \geq (3t-1)(t-1)$ . Our approach utilizes a connection to the related problem of determining the maximum of the minimum degrees among the family of balanced $r$ -partite $rn$ -vertex graphs of chromatic number at most $t$ .

Measurement, modeling, reduction of decision-theoretic multigranulation fuzzy rough sets based on three-way decisions

Variable precision multigranulation fuzzy rough sets (VP-MFRSs) use two direct integrations: the multigranulation maximum and minimum. Their optimistic and pessimistic models facilitate uncertainty informatization but also imply a potential limitation of extremes. This study improves VP-MFRSs for extension and balance, and thus, decision-theoretic multigranulation fuzzy rough sets (DT-MFRSs) are proposed by systematically fusing the multigranulation maximum and minimum. For DT-MFRSs, their tri-level analysis of measurement, modeling, and reduction is deeply acquired via three-way decisions. First, maximum and minimum membership degrees are linearly combined, and the weight parameter guides a generalized multigranulation membership degree. This adjustable measure motivates DT-MFRSs with positive, negative, and boundary regions, while attitude-preference values of 1, 0, and 0.5 respectively produce optimistic, pessimistic, and compromised models. Then, nonmonotonicity and uncertainty of membership degrees and model regions are determined, and these fundamental characteristics induce new reduction criteria of region preservations. Also, three-way attribute reducts are proposed by preserving positive, negative, and positive–negative regions, and their systematic relationships are obtained. Finally, tri-level results of measures, models, and reducts are validated by table examples and data experiments. In this study, DT-MFRSs extend and improve VP-MFRSs via systematic fusion of membership measurement, and contain optimistic, pessimistic, compromised models, etc., thereby exhibiting extended diversity and applied robustness. Their three-way attribute reduction also perfects uncertainty optimization.

Topological aspects of extended Sierpiński structures with help of underlying networks

Sierpiński networks are the most studied networks of fractal nature with applications in various fields of science. A generalized Sierpiński network is obtained by copying the base network, resulting in the self-similar network. The extended Sierpiński networks are obtained by introducing a new vertex in a generalized Sierpiński network and attaching this vertex with the extreme vertices. Certain network invariants are used to find thermodynamic properties, physio-chemical properties, and biological activities of chemical compounds. These network invariants play a dynamic role in QSAR/QSPR study. In this paper, we discussed Zagreb indices and forgotten topological index for extended Sierpiński networks by using any base network H. Moreover, for the studied topological indices, we attained some bounds using different parameters i.e. order, size, maximum and minimum degrees of vertices in network H.

Minimum Levels of Human Intervention in Autonomous Attacks

Abstract This article discusses an important limitation on the degree of autonomy that may permissibly be afforded to autonomous weapon systems (AWS) in the context of an armed conflict: the extent to which international humanitarian law (IHL) requires that human beings be able to intervene directly in the operation of weapon systems in the course of an attack. As there is currently no conventional or customary law directed specifically at AWS, limits on use of autonomous capabilities in weapons, if any exist, must be inferred from the principles, rules and goals of general IHL. The process adopted herein is to look for two broad types of limitations: those which take the form of maximum permissible degrees of machine involvement in regulated activities, and those which take the form of minimum permissible degrees of human involvement. The article’s main finding is that while existing law does not impose limits of the first type, it does impose some of the second type. Specifically, legal obligations borne by individuals (commanders in charge of AWS operations, weapon system operators and others) determine the required minimum capacity for direct human intervention. The article further suggests means by which the required degree of human intervention may be determined in specific circumstances.

NeuroMechFly, a neuromechanical model of adult Drosophila melanogaster.

Animal behavior emerges from an interaction between neural network dynamics, musculoskeletal properties and the physical environment. Accessing and understanding the interplay between these elements requires the development of integrative and morphologically realistic neuromechanical simulations. Here we present NeuroMechFly, a data-driven model of the widely studied organism, Drosophila melanogaster. NeuroMechFly combines four independent computational modules: a physics-based simulation environment, a biomechanical exoskeleton, muscle models and neural network controllers. To enable use cases, we first define the minimum degrees of freedom of the leg from real three-dimensional kinematic measurements during walking and grooming. Then, we show how, by replaying these behaviors in the simulator, one can predict otherwise unmeasured torques and contact forces. Finally, we leverage NeuroMechFly's full neuromechanical capacity to discover neural networks and muscle parameters that drive locomotor gaits optimized for speed and stability. Thus, NeuroMechFly can increase our understanding of how behaviors emerge from interactions between complex neuromechanical systems and their physical surroundings.

General Atom-Bond-Connectivity Index Of Graphs

The Atom-bond-connectivity index ABC of a graph G is determined by d_i and d_j. In this paper, sharp results for the general ABC index which has chemical applications are found using different methods. These new results for ABC inex are investigated in terms of its edges, its vertices and its degrees. In particular, some relations for general ABC index is obtained involving different topological indices; Randic index, Zagreb index, Harmonic index and Narumi-Katayama index. Indeed, general ABC index are improved by the help of the maximum annd minimum degrees.

On the asymmetric generalizations of two extremal questions on friends-and-strangers graphs

For two graphs X and Y with vertex sets V(X) and V(Y) of the same cardinality n, the friends-and-strangers graph FS(X,Y) was recently defined by Defant and Kravitz. The vertices of FS(X,Y) are the bijections from V(X) to V(Y), and two bijections σ and τ are adjacent if they agree everywhere except at two vertices a,b∈V(X) such that a and b are adjacent in X and σ(a) and σ(b) are adjacent in Y. We study generalized versions of two problems by Alon, Defant, and Kravitz. First, we show that if X and Y have minimum degrees δ(X) and δ(Y) that satisfy δ(X)>n/2,δ(Y)>n/2, and 2min(δ(X),δ(Y))+3max(δ(X),δ(Y))≥3n, then FS(X,Y) is connected. As a corollary, we settle a recent conjecture by Alon, Defant, and Kravitz stating that there exists a number dn=3n/5+O(1) such that if both X and Y have minimum degrees at least dn, the graph FS(X,Y) is connected. When X and Y are bipartite, a parity obstruction prevents FS(X,Y) from being connected. We show that if X and Y are edge-subgraphs of Kr,r that satisfy δ(X)+δ(Y)≥3r/2+1, then the graph FS(X,Y) has exactly two connected components. As a corollary, we provide an almost complete answer to another recent question of Alon, Defant, and Kravitz asking for the minimum number dr,r∗ such that for any edge-subgraph X of Kr,r satisfying δ(X)≥dr,r∗, the graph FS(X,Kr,r) has exactly two connected components. We show that dr,r∗=r/2+1 when r is even and dr,r∗∈{⌈r/2⌉,⌈r/2⌉+1} when r is odd.

Graph Indices for Cartesian Product of -sum of Connected Graphs.

A topological index is a real number associated with a graph that provides information about its physical and chemical properties and their correlations. Topological indices are being used successfully in Chemistry, Computer Science, and many other fields. In this article, we apply the well-known Cartesian product on F-sums of connected and finite graphs. We formulate sharp limits for some famous degree-dependent indices. Zagreb indices for the graph operations T(G), Q(G), S(G), R(G), and their F-sums have been computed. By using orders and sizes of component graphs, we derive bounds for Zagreb indices, F-index, and Narumi-Katayana index. The formulation of expressions for the complicated products on F-sums, in terms of simple parameters like maximum and minimum degrees of basic graphs, reduces the computational complexities.

Hall-Higman type theorems for exceptional groups of Lie type, I

The paper studies the minimum polynomial degrees of p-elements in cross-characteristic representations of simple groups of exceptional Lie type whose BN-pair rank is at most 2. Specifically, we prove that the degree in question equals the order of the element.

Fuzzy Association Rules Mining based on Type-2 Fuzzy Sets over Data Stream

Fuzzy set theory aims to solve the problem of mining more valuable human reasoning knowledge from quantitative databases, and to narrow the gap between computer and human understanding. Fuzzy Frequent Itemset (FFI) mining, which is very important in fuzzy association rule mining, converts quantitative values into language, and finds fuzzy frequency itemset according to the defined membership function. In the traditional fuzzy frequent pattern mining method, only the language item with the largest scalar base is considered, and the uncertainty of the membership function is not considered. At the same time, it is mainly considered for the fixed data set. This paper proposes a Fuzzy Frequent Pattern mining algorithm based on the Type-2 Fuzzy Set (T2FS) theory of the data stream. Under the T2FS theory, the data stream is dynamically divided based on the sliding window method, and the ambiguity is quickly found from the numerical data stream. In order to reduce the search space and improve over-mining, this paper designs a dynamic compressed lists structure and trees method. Under different minimum support degrees and sliding window sizes, many experiments have been carried out, and the experiments have verified the efficiency and effectiveness of the designed method in terms of running time and memory usage. The results can also contribute to exchange the information and preferences among the people who cooperate in decision-making.

A general framework for sliding cable analysis with elastic catenary equation

Sliding cables are widespread in engineering applications. Most of the existing sliding cable models are developed basing on conventional straight truss element, which is not applicable for many engineering applications where structures possess long-span and large sag effects. To effectively analyze this kind of system, this paper proposes a multi-node sliding catenary cable element. Due to the usage of analytic form of elastic catenary equation, the proposed element has the capability to accurately analyze both very taut and very sagged cables with minimum degrees of freedom. The formulation is developed within both dynamic relaxation approach and fully implicit approach. Effective numerical strategies in the nonlinear iteration, as well as the related tangent matrices are rigorously given. In particular, the multi-loop strategy permits to integrate the very complex nonlinear element into general finite element procedure, without needing to introduce any additional variable in the system. Multiple examples are presented to demonstrate the validity of the proposed framework. The numerical results also emphasize that sliding effects have a significate influence on the mechanical behavior of the catenary cable system, while can be easily addressed by using the methodology developed in this work.

Zero or minimum degree? Rethinking minimum gradable adjectives

This paper examines the class of minimum gradable adjectives (minimum GAs), the standards of whose positive forms are traditionally characterized in terms of minimum degrees. Using profitable as a case study, I argue that their standards are better characterized using zero degrees, which may not be minimum degrees. Moreover, I show that this zero-standard interpretation cannot be derived by existing approaches, and propose that it is derived by a newcompositional mechanism, which also derives the truth conditions of plain comparatives and is different from the one that derives the relative-standard interpretation of a positive form.

Lower bounds for the [formula omitted]-spectral radius of uniform hypergraphs

For 0≤α<1, the Aα-spectral radius of a k-uniform hypergraph G is defined to be the spectral radius of the tensor Aα(G):=αD(G)+(1−α)A(G), where D(G) and A(G) are the diagonal and adjacency tensors of G, respectively. This paper presents several lower bounds for the difference between the Aα-spectral radius and an average degree kmn of a connected k-uniform hypergraph G with n vertices and m edges, which are considered as measures of irregularity of G. Moreover, two lower bounds on the Aα-spectral radius are obtained in terms of the maximum and minimum degrees of uniform hypergraphs.

Combinatorial generation via permutation languages. II. Lattice congruences

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.

Bounds for the Energy of Graphs

Let G be a graph on n vertices and m edges, with maximum degree Δ(G) and minimum degree δ(G). Let A be the adjacency matrix of G, and let λ1≥λ2≥…≥λn be the eigenvalues of G. The energy of G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G, that is E(G)=|λ1|+…+|λn|. The energy of G is known to be at least twice the minimum degree of G, E(G)≥2δ(G). Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G, i.e., E(G)≥Δ(G)+δ(G). In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E(G). The results rely on elementary inequalities and their application.