A Quantum Walk Model for Idea Propagation in Social Network and Group Decision Making

We propose a quantum walk model to investigate the propagation of ideas in a network and the formation of agreement in group decision making. In more detail, we consider two different graphs describing the connections of agents in the network: the line graph and the ring graph. Our main interest is to deduce the dynamics for such propagation, and to investigate the influence of compliance of the agents and graph structure on the decision time and the final decision. The methodology is based on the use of control-U gates in quantum computing. The original state of the network is used as controller and its mirrored state is used as target. The state of the quantum walk is the tensor product of the original state and the mirror state. In this way, the proposed quantum walk model is able to describe asymmetric influence between agents.

On Efficient Zero Ring Labeling and Restricted Zero Ring Graphs

In [3], Acharya et al. introduced the notion of a zero ring labeling of a connectedgraph G, where vertices are labeled by the elements of a zero ring such that the sum of the labels of adjacent vertices is not the additive identity of the ring. Archarya and Pranjali [1] also constructed a graph based on a finite zero ring called the zero ring graph. In [5], Chua et al. defined a class of zero ring labeling called efficient zero ring labeling and it was shown that a labeling scheme exists for some families of trees. In this paper, we provide an efficient zero ring labeling for some classes of graphs. We also introduce the notion of the restricted zero ring graphs and use them to show that a zero ring labeling exists for some classes of cactus graphs.

Max-independent set and the quantum alternating operator ansatz

The maximum-independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, [Formula: see text], and [Formula: see text] graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum-independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs, it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. We also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.

d-Path Laplacians and Quantum Transport on Graphs

We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph—a graph consisting of two cliques separated by a path—the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d-path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d-path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian.

When the maximal graph is planar, outerplanar, and ring graph

Let [Formula: see text] be a commutative ring with nonzero identity. Let [Formula: see text] denote the maximal graph associated to [Formula: see text], that is, [Formula: see text] is a graph with vertices as non-units of [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there is a maximal ideal of [Formula: see text] containing both. In this paper, we characterize the finite commutative rings such that their maximal graph are planar graphs, and we also study the case where they are outerplanar and ring graphs. The equivalence of outerplanar graphs and ring graphs for [Formula: see text] is established.

A Combinatorial Approach for Developing Ring Communication Graphs for Vehicle Formations

In this paper, we study vehicle formations employing ring-structured communication strategies and propose a combinatorial approach for developing ring graphs for vehicle formations. In vehicle platoons, a ring graph is formed when each vehicle receives information from its predecessor, and the lead vehicle receives information from the last vehicle, thus forming a ring in its basic form. In such basic form, the communication distance between the first and the last vehicle increases with the platoon size, which creates implementation issues due to sensing range limitations. If one were to employ a communication protocol such as the token ring protocol, the delay in updating information and communication arises from the need for the token to travel across the entire graph. To overcome this limitation, alternative ring graphs which are formed by smaller communication distances between vehicles are proposed in this paper. For a given formation and a constraint on the maximum communication distance between any two vehicles, an algorithm to generate a ring graph is obtained by formulating the problem as an instance of the traveling salesman problem (TSP). In contrast to the vehicle platoons, generation of a ring communication graph is not straightforward for two- and three-dimensional formations; the TSP formulation allows this for both two- and three-dimensional formations with specific constraints. In addition, with ring communication structure, it is possible to devise simple ways to reconfigure the graph when vehicles are added/removed to/from the formation, which is discussed in the paper. Further, the experimental results using mobile robots for platooning and two-dimensional formations using ring graphs are shown and discussed.

Crystal structure of [1-(3-ethoxy-2-oxidobenzylidene-κO2)-4-phenylthiosemicarbazidato-κ2N1,S](triphenylphosphane-κP)nickel(II)

In the title complex, [Ni(C16H15N3O2S)(C18H15P)], the NiIIatom has a distorted tetrahedral coordination geometry, comprised of N, S, O and P atoms of the tridentate thiosemicarbazide ligand and the P atom of the triphenylphosphane ligand. The benzene ring makes a dihedral angle of 53.08 (11)° with the phenyl ring of the phenylthiosemicarbazide moiety and dihedral angles of 73.69 (11), 20.38 (11) and 71.30 (11)° with the phenyl rings of triphenylphosphane ligand. A pair of N—H...N hydrogen bonds generates anR22(8) ring graph-set motif. The ethoxy group is disordered over two positions, with site occupancies of 0.631 (9) and 0.369 (9). The molecular structure is stabilized by a weak intramolecular C—H...O hydrogen bond. In the crystal, weak N—H...N and C—H...π interactions connect the molecules, forming a three-dimensional network.

Graphs and complete intersection toric ideals

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal PG is a complete intersection or not. Whenever PG is a complete intersection, the algorithm also returns a minimal set of generators of PG. Moreover, we prove that if G is a connected graph and PG is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

Research on Manufacturing of Blade Screw in Solid-Liquid Separator

Blade screw propeller is used as the pushing and squeezing machine which is often used in the liquid separator for software materials. The core part of blade screw is stainless steel material with wide tooth, thin wall and other characteristics, and its processing method is different from the cylindrical surface spiral groove processing. Usually, its machining process is based on a certain forming technology to produce spiral leaf firstly, then welding leaves on the shaft. The cylindrical blade screw in test equipment adopts triangle method to calculate the blank dimensions. Variable cross section of the outer ring graph is realized by using the polar radius equation drawing in the drawing software.

CAYLEY SUM GRAPHS OF IDEALS OF A COMMUTATIVE RING

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a commutative ring, $I(R)$ be the set of all ideals of $R$ and $S$ be a subset of $I^*(R)=I(R)\setminus \{0\}$. We define a Cayley sum digraph of ideals of $R$, denoted by $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$, as a directed graph whose vertex set is the set $I(R)$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$, denoted by $I\longrightarrow J$, whenever $I+K=J$, for some ideal $K $ in $S$. Also, the Cayley sum graph $ \mathrm{Cay}^+ (I(R), S)$ is an undirected graph whose vertex set is the set $I(R)$ and two distinct vertices $I$ and $J$ are adjacent whenever $I+K=J$ or $J+K=I$, for some ideal $K $ in $ S$. In this paper, we study some basic properties of the graphs $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$ and $ \mathrm{Cay}^+ (I(R), S)$ such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of $ \mathrm{Cay}^+ (I(R), S)$ and also we provide some characterization for rings $R$ whose Cayley sum graphs have genus one.